i^lCWRLF 


IL^SO  SOI 


NUMERICAL  PROBLEMS 

IN 

PLANE  GEOMETRY 


J  «    VJr   .  t  ^   I    I  LiL 


LIBRARY 

OF  THE 

University  of  California. 


f 


GIFT    OF 


..>?...:.....VSr:.:. .yv:V..9vA<\^Jo^^ 

Class 


9vt4^  .f /.£ 


NUMEEIOAL  PEOBLEMS 

IN 

PLANE  GEOMETRY 


NUMERICAL    PROBLEMS 


m 


PLANE  GEOMETRY 


METRIC  AND  L0GAE1TH3IIC  TABLES 


BY 

J.  G.  ESTILL 

OF  THE  HOTCHKISS  SCHOOL,   T.AKEVILLE,   CONN. 


NEW    YORK 
LONGMANS,   GEEEN,   AND    CO. 

LONDON  AND  BOMBAY 

1897 


Copyright,  1896 

BY 

LONGMANS,   GKEEN,  Ax\D  CO. 


MANHATTAN  PRESS 

474  W.  BROADWAY 

NEW  YORK 


PREFATORY  NOTE 


When^  arithmetic  was  dropped  from  the  requirements 
for  admission  to  Yale  College,  in  1894,  the  following  sub- 
stitute was  adopted  :  '^  Plane  Geometry  (b) — Solution  of 
numerical  problems  involving  the  metric  system  and  the 
use  of  Logarithms,  also  as  much  of  the  theory  of  Loga- 
rithms as  is  necessary  to  explain  their  use  in  simple  arith- 
metical operations. — Five-figure  tables  will  be  used  in  the 
examination/'     (1896-97  Catalogue.) 

At  the  conference  on  uniform  requirements  for  admis- 
sion to  college,  in  February,  1896,  at  Columbia  College, 
representing  Harvard,  Yale,  Princeton,  University  of 
Pennsylvania,  Columbia,  and  Cornell,  and  nearly  all  the 
large  preparatory  schools  of  the  East,  the  Mathematical 
Conference  voted  unanimously  to  recommend  that  arith- 
metic be  dropped  from  the  college  entrance  requirements, 
and  that  a  knowledge  of  the  metric  system  and  the  abil- 
ity to  solve  numerical  problems  in  Plane  Geometry  be 
required. 

These  two  facts  account  for  the  writing  of  this  little 
book. 

The  most  of  the  problems  have  had  class-room  test. 
They  add  interest  to  the  study  of  formal  geometry.  They 
are  helpful,  too,  in  making  clear,  and  fastening  in  the 
memory,  the  principles  and  propositions  of  formal  geom- 
etry.     They  enforce  the  practical  application  of  truths 


183647. 


VI  PREFATOBT  NOTE. 

which  boys  are  apt  to  think  have  no  application.  They 
furnish  a  drill  that  is  just  as  valuable  to  those  who  are  not 
preparing  for  college  as  for  those  who  are.  These  prob- 
lems are  not  to  take  the  place  of  other  geometries,  but  are 
to  be  used  with  them.  And,  therefore,  the  division  into 
Books  is  made  to  correspond  pretty  closely  with  that  of  the 
geometries  in  most  general  use. 

The  use  of  the  metric  system  is  begun  at  the  very  first, 
simple  as  that  necessarily  makes  the  problems  of  the  first 
book,  for  the  most  part.  No  other  book  contains  a  graded 
set  of  problems  on  the  first  two  books  of  geometry. 

No  apology  is  considered  necessary  for  putting  in  quite 
a  number  of  problems  which  presuppose  some  knowledge 
of  algebra. 

The  order  of  the  problems  is  not  the  same  as  the  order 
of  the  propositions  of  any  geometry  ;  neither  are  all  the 
problems  which  illustrate  an  important  principle  placed 
together.  The  reason  for  this  is  obvious.  Still,  the  order 
of  the  problems  in  the  different  books  is  approximately  the 
same  as  the  order  of  the  propositions  in  the  most  popular 
text-books.  On  account  of  this  difference  in  order  it  will 
be  best  to  keep  the  text-book  work  somewhat  ahead,  unless 
one  cares  to  select  the  problems  beforehand  to  give  out 
with  the  text-book  lesson.  Some  may  prefer  to  use  the' 
problems  only  with  the  review  of  the  geometry. 

Boys  preparing  for  college  will  certainly  take  a  lively  in- 
terest in  the  questions,  problems,  and  exercises  selected 
from  the  college  entrance  papers. 

The  entrance  papers  were  selected  with  great  care,  with 
the  hope  that  they  may  prove  helpfully  suggestive  both  to 
teachers  and  pupils. 

The  discussion  of  logarithms,  the  explanation  of  their 
use,  and  the  use  of  the  table  have  been  made  as  simple 
and  clear  as  possible. 


PREFATORY  NOTE.  idi 

Only  such  symbols  are  used  as  are  almost  universally 
employed. 

Some  few  proofs  are  put  in  because  they  are  not  found 
in  all  the  text-books. 

Notice  of  errors,  or  any  suggestion,  will  be  gratefully  re- 
ceived. 

J.  G.  ESTILL. 
HoTCHKiss  School, 
Lakeyille,  Conn.,  January  8,  1897. 


CONTENTS 

PAGB 

Preface      v 

Book  I .        „  1 

Book  II .  11 

Book  III 17 

Book  IV.             29 

Book  V 40 

Numerical  Problems,  Exercises,  Etc., 
Selected  from  Entrance  Examina- 
tion Papers 50 

College  Examination  Papers  in  Plane 

Geometry 66 

Logarithms 102 

Examples Ill 

Tables 115 


NUMERICAL   PROBLEMS 


IN 


PLANE   GEOMETRY 


BOOK  I. 

1.  What  is  the  complement  of  43°  ?  of  75°  15'  ?  of  81° 
11'  ir  ?  of  14°  18"  ?  of  ii  [^  ?  of  m°  n'  ?  of  82°  40'  -  .4 

2.  What  is  the  supplement  of  28°  31'  18"  ?  of  115°  39"  ? 
of  140°  1.84"  ?  of  1.2  L^  ?  of  tV  Ll  ?  of  c°-  f"  ? 

3.  Find  the  supplement  of  the  complement  of  50° ;  of 
85°  13'  22"  ;  of  a;°  ;  of  ^5°  -  31°  18'. 

4.  Find  the  complement  of  the  supplement  of  169°  44' 
42"  ;  of  155°  55"  ;  of  ^°  -  15°  ;  of  c°  -  8°  5". 

5.  How  many  degrees  in  the  difference  between  the 
supplement  and  the  complement  of  an  /  ? 

6.  How  many  degrees  in  each  of  the  A  made  by  two 
intersecting  straight  lines,  when  one  of  the  A  lacks  only 
2°  of  being  }  of  J  of  a  [^  ? 

7.  In  this  figure  ,_\y3,  ^  =  /,  =  ^  +  12° ;  how 
many  degrees  in  each  /_  ? 


2  OEOMETBY— NUMERICAL  PROBLEMS, 

8.  Of  two  supplementary-adjacent  A  ,  one  lacks  7°  of 
being  ten  times  as  large  as  the  other ;  how  many  degrees 
in  each  ? 

9.  Two  complementary  A  are  such  that  if  7°  be  added 
to  one  and  8°  to  the  other  they  will  be  in  the  ratio  of 

3  to  4? 

10.  If  an  /  divided  by  its  supplement  gives  a  quotient 
of  5  and  a  remainder  of  6°,  how  many  degrees  in  the  /  ? 

11.  How  many  degrees  in  each  of  the  five  A  about  a 
point,  if  each,  in  a  circuit  from  right  to  left  is  5°  greater 
than  its  adjacent  /  ? 

12.  Three  A  make  up  all  the  angular  magnitude  about 
a  point.  The  difference  between  the  first  and  second  is 
10°  ;  the  difference  between  the  second  and  third  is  100°  ; 
how  many  degrees  in  each  ? 

13.  When  the  A  formed  by  one  straight  line  meeting 
another  are  in  the  ratio  7  :  11  how  many  degrees  in  each  ? 

14.  Find  the  /  whose  complement  and  supplement  are 
in  the  ratio  4  :  13. 

15.  Find  the  /  the  sum  of  whose  supplement  and  com- 
plement is  15°  less  than  four  times  its  complement. 

16.  How  many  degrees  in  the  /  whose  supplement 
taken  from  three  times  its  complement  leaves  1°  18'  less 
than  the  difference  between  the  /  and  50°  ? 

17.  If  the  bisector  of  one  of  two  supplementary-adjacent 
A  makes  an  /  equal  to  one-sixth  of  the  other,  how  many 
degrees  in  each  of  the  A  ? 

18.  How  many  degrees  in  each  of  the  five  A  about  a 
point  if  they  are  in  the  ratio  1:2:3:4:5? 


GEOMETRY— NUMERICAL  PROBLEMS.  3 

19.  What  answer  to  18  if  the  ratio  is  3  :  3  :  7  :  11  :  13  ? 

20.  If  the  complement  of  the  /  A  is  three  and  one-half 
times  as  large  as  A,  what  part  of  7  |_^  is  the  /  A  ? 

21.  Find  the  /  whose  supplement  increased  by  26°  will 
be  three  times  its  complement. 

22.  How  many  degrees  in  the  /  whose  supplement  and 
complement  added  together  make  144°  ? 

23.  How  many  degrees  in  the  /  whose  supplement,  in- 
creased by  9°,  is  to  its  complement,  decreased  by  1°,  as  7 
to  2? 

24.  Find  the  number  of  degrees  in  each  of  these  A,    a  c/ 
if  h  is  2°  less  than  f  of  «  ;  c  is  f  (o^  +  ^  —  1°) ;  ^  is  ^t^^^ 
13°  less  than  the  sum  of  a,  h,  and  c ;  and  e  is  2° 

more  than  the  difference  between  the  sum  of  h  and  d,  and 
the  sum  of  a  and  c. 

25.  How  many  degrees  in  the  /  whose  complement  is 
one-fifth  its  supplement  ? 

26.  How  many  degrees  in  the  /  whose  supplement,  in- 
creased by  20°,  divided  by  its  complement,  decreased  by 
5°,  gives  a  quotient  4  and  a  remainder  25°  ? 

27.  If  a  J_  is  1  foot  10  inches  from  one  end  of  a  line 
and  SS*""  from  the  other,  at  what  point  of  the  line  is 
this  1  ? 

28.  Of  two  lines  from  the  same  point  to  the  same 
straight  line,  one  is  1  yard  1  foot  4  inches,  the  other  is 
130*"°,  what  can  you  say  of  them  ? 

29.  Two  lines  from  a  point  to  the  extremities  of  a 
straight  line  are  15  feet  4  inches,  and  11  feet  11  inches. 


V 


4  GEOMETRY— NUMERIGAL  PROBLEMS. 

respectively.     Two  similarly  drawn  are  4™  ^^"^  and  3.2™. 
Which  pair  includes  the  other  ?    Why  ? 

30.  Of  two  oblique  lines  from  a  point  to  a  straight  line 
one  is  3  feet  10.8  inches,  the  other,  1™  1*^'°  7<^™ ;  which  cuts 
off  the  greater  distance  from  the  foot  of  the  perpendicular 
from  the  point  to  the  straight  line. 

31.  What  answer  to  30,  if  the  lines  are  35  feet  and  1^°*, 
respectively  ? 

32.  If  the  bisector  of  one  of  two  supplementary-adjacent 
A  makes  with  their  common  side  an  /  =  |  |_r_  lacking 
6°,  how  many  degrees  in  the  other  /  ? 

33.  Of  two  lines  from  a  point  to  a  straight  line,  one  is 
30*"°  and  the  other  is  11  inches,  which  is  a  _L,  if  either  is  ? 
Why? 

34.  Which  is  the  greater  of  two  oblique  lines  from  a  point 
to  a  straight  line,  cutting  off,  the  one  20  yards,  the  other 
15",  from  the  foot  of  the  1  from  the  point  to  the  line  ? 

35.  Answer  the  same  when  the  distances  cut  off  are  1" 
^dm  5cm  and  5  feet  10  inches. 

36.  In  the  A  A  B  C  and  A'  B'  C,  «  =  3  feet,  5  =  7 
feet,  c  =  8  feet,  /  A  =  /_M,  V  =1  feet,  c'  =  8  feet. 
Find  the  length  of  a'  in  centimetres.* 

37.  In  the  A  A  B  C,  «  =  4",  J  =  5™,  c  =  7°* ;  find  in 
feet  (approximately)  the  sides  of  a  A  eq^al  to  the  /\ 
ABO. 

38.  One  side  of  a  A  is  1"  5^™,  another  7  feet  5  inches. 
What  is  the  greatest  value  the  third  side  can  have  (1)  in 
metric  units,  (2)  in  English  units  ?     What  is  the  least  ? 

♦  a,  6,  c,  represent  the  Sides  of  a  A  opposite  the  A  A,  B,  C,  re- 
spectively. 


QEOMETRT— NUMERICAL  PROBLEMS.  5 

39.  Find  the  ^  of  the  /\  A  B  0,  when  A  is  43°  more 
than  I  of  B,  which  is  18°  less  than  4  times  0. 

40.  In  the  two  A  A  B  0  and  A'  B'  C,  A  =  37°,  B  = 
111°,  c  =  2.5  feet,  A'  =  111°,  B'  =  37°,  d  =  7'^"  S^-". 
What  can  you  say  of  them  ?    Why  ? 

41.  In  the  A  A  B  0,  «  =  13  feet,  ^>  =  17.3  feet,  and  c  = 
22. 4  feet,  find  in  metres  (approximately)  the  sides  of  a  A  = 
the  A  A  B  C.     (Log.*) 

42.  One  of  the  acute  zi  of  a  right  A  =  37°  and  the  hy- 
potenuse is  1.5  miles,  how  many  kilometres  in  the  hypo- 
tenuse of  an  equal  right  A  which  has  an  acute  /  of  37°  ? 

43.  In  the  A  A  B  C,  «  =  11^^  h  =  32'^'",  what  is  the 
least  possible  value  in  miles  of  the  side  c  ? 

44.  If  in  two  A  A  B  C  and  A'  B'  C,  a  =  1^  5"^",  b  = 
1"  2'^'°  S'^'",  C  =  48°,  a'  =  3  feet  6  inches,  Z»'  =  4  feet 
2  inches,  C  =  148°,  what  can  you  say  of  c  and  (/  ?  Show 
by  your  work  how  you  reached  your  conclusion. 

What  would  your  answer  be  if  all  the  given  values  were 
the  same  except  C  =  48°  ?    Why  ? 

45.  If  in  two  A  A  B  C  and  A'  B'  C,  «  =  7  miles,  b  = 
13  miles,  c  -  15  miles,  a'  =  lli''^  b'  =  21^'^  c'  =  24^"", 
what  about  the  A  B  and  B'  ?  If  ^>'  =  201^^  what  of  these 
A? 

46.  In  the  A  A  B  0,  a  =  1.3  miles  and  b  =  2^°^,  what 
of  the  A  A  and  B  ?  It  a  were  the  same  and  b  =  2.08^"*, 
what  could  you  say  of  the  ^  A  and  B  ? 

47.  The  ^  A  and  B  in  the  A  A  B  C  are  each  49°  18' 

♦  Certain  problems  m  each  book  are  marked  thus  for  those  who  care 
for  practice  in  the  use  of  logarithms. 


6  OEOMETRT— NUMERICAL  PROBLEMS. 

and  a  —  109  yards  1  foot  1  inch,  how  many  metres  in  the 
side  b  ?  (Log.) 

48.  If  one  of  the  A  made  by  a  line  cutting  two  |I  lines 
is  3°  more  than  y\  |j^ ,  how  many  degrees  in  each  of  the 
other  A  ?    (Mark  your  answers  on  a  figure. ) 

49.  What  answer  to  48  if  one  of  the  A  is  eight  times 
its  conjugate  /  ? 

50.  If  the  exterior  /  at  A  of  the  A  ^  B  C  is  115°,  and 
/  C  is  three  times  /  B,  find  B  and  C. 

51.  The  exterior  A  at  A  and  0  of  the  A  ^  B  C  are  71° 
and  92°  respectively ;  how  many  degrees  in  the  /  B  ? 

52.  In  the  A  A  B  C,  A  lacks  106°  of  being  equal  to  the 
sum  of  B  and  C,  and  C  lacks  10°  of  being  equal  to  the 
sum  of  A  and  B  ;  find  A,  B,  and  C. 

53.  Find  the  ^  of  a  /\  which  are  in  the  ratio  3:4:5. 

54.  Find  the  A  of  an  isosceles  /\  in  which  the  exterior 
/  at  the  vertex  is  125°. 

55.  Find  the  A  of  an  isosceles  /^  in  which  the  exterior 
/  at  the  base  is  95°. 

56.  Find  the  perimeter  of  an  isosceles  /\,  in  miles,  if  a 
base  of  48'^'°  is  the  longest  side  of  the  /\  by  12^"".   (Log.) 

57.  In  the  A  A  B  0,  a  =  15  yards  and  h  =  1^^  2"",  what 
about  the  A  A  and  B  ? 

58.  The  point  P  in  the  bisector  of  the  angle     ^<^  V — 

is  5  yards  2  feet  from  the  side  1—2-,  how  many  metres  is 
P  from  2-3  ? 


GEOMETRY— NUMERICAL  PROBLEMS.  7 

59.  The  point  P  within  an  /  is  6  <^™  6^^  from  one  side 
of  the  /  and  2  feet  2  inches  from  the  other  side,  where 
does  it  lie  ?  Show  the  reason  for  your  answer  by  your 
work. 

60.  The  /  at  the  vertex  of  an  isosceles  ^  is  one-third 
the  exterior  angle  at  the  vertex,  how  many  degrees  in  each 
/ ,  exterior  and  interior,  at  the  base  ? 

61.  In  the  A  A  B  C,  A  =  35°,  B  =  45°,  a  =  J  mile  ; 
what  can  you  say  of  the  length  of  h,  in  metres  ? 

62.  Two  adjacent  sides  of  a  /  /  are  respectively  18"  and 
21™  ;  find  the  lengths  of  the  other  two  sides  in  yards.    (Log. ) 

63.  The  area  of  one  of  the  A  made  by  the  diagonal  of  a 
/      /  is  5.2"*.     How  many  acres  in  the  other  ? 

64.  If  one  /  of  a  /  /  =  J  L^ ,  how  many  degrees  in 
each  of  the  other  A  ? 

65.  If  two  adjacent  zi  of  a  /  /  are  in  the  ratio  of 
17  : 1,  how  many  degrees  are   there  in  each    /   of  the 

66.  How  many  degrees  in  each  /  of  a  /  /  where  one 
/  exceeds  one-third  of  its  adjacent  /  by  two-thirds  of  a 
degree  ? 

67.  How  many  degrees  in  each  /  of  an  equiangular 
icosagon  ?  in  each  exterior  /  ? 

68.  How  many  sides  has  the  polygon  each  of  whose  ex- 
terior ^  =  12**  ? 

69.  How  many  sides  to  the  polygon  each  of  whose  ex- 
terior A  is  only  one-eleventh  of  its  adjacent  interior  /  ? 

70.  One  side  of  a  rhombus  is  13.6/°  find  its  perimeter 
in  miles.     (Log.) 


8  OEOMETRT—NUMERIGAL  PROBLEMS. 

71.  One  side  of  a  rhomboid  is  4  feet  longer  than  the 
other,  the  perimeter  is  14",  what  are  the  lengths  of  the 
sides  in  feet  and  inches  ?     (Log.) 

72.  Find  the  number  of  acres  in  a  rhombus  in  which 
one  of  the  four  A  made  by  the  diagonals  contains  5.11"*. 
(Log.) 

73.  'Find  the  A  of  an  isosceles  l\  when  one  of  the  A  at 
the  base  is  equal  to  one-half  the  /  at  the  vertex. 

74.  What  answer  to  73,  when  the  /  at  the  vertex  is  9° 
greater  than  an  /  at  the  base  ? 

75.  What  are  the  A  of  an  isosceles  /\  in  which  the  / 
at  the  vertex  is  12**  more  than  one-third  the  sum  of  the 
base  A't  ^ 

76.  The  sides  of  a  quadrilateral  taken  in  order  are  6 
inches,  18^™,  15«",  7^  inches,  respectively.  What  is  the 
nature  of  this  quadrilateral  ? 

77.  How  many  sides  has  the  polygon  each  of  whose  in- 
terior A  =  171°  ? 

78.  The  line  joining  the  middle  points  of  two  sides  of  a 
/\  is  2.5  miles,  what  is  the  length  of  the  third  side  in  kilo- 
metres ? 

79.  How  many  sides  has  the  polygon  the  sum  of  whose 
interior  A  exceeds  the  sum  of  its  exterior  A  by  3240°  ? 

80.  One  of  the  diagonals  of  a  rectangle  is  40  yards  2  feet 
10  inches  ;  find  the  length  of  the  other  in  metres.     (Log.) 

81.  One  base  of  a  trapezoid  is  125°™,  the  line  joining  the 
middle  points  of  the  non-parallel  sides  .  7™,  find  the  length 
of  the  other  base. 

82.  How  many  sides  has  the  equiangular  polygon  each 
of  whose  interior  A  exceeds  its  adjacent  exterior  by  108°  ? 


QEOMETBT— NUMERICAL  PROBLEMS.  9 

83.  How  many  sides  has  the  polygon  the  sum  of  whose 
interior  A  is  double  the  sum  of  the  exterior  A  ? 

84.  The  line  joining  the  middle  points  of  the  non-paral- 
lel sides  of  a  trapezoid  is  13  feet  5  inches,  and  one  of  the 
bases  is  2J  times  as  long  as  the  other  ;  find  the  length  of 
the  bases. 

85.  Find  the  length  in  metres  of  the  line  which  bisects 
one  side  of  a  /\  and  is  parallel  to  a  side  whose  length  is  9 
feet  10.11  inches. 

86.  If  you  should  join  the  extremities  of  two  parallel 
lines  whose  lengths  are  7^"  and  4.375  miles  respectively, 
what  kind  of  a  figure  would  be  formed  ?    Why  ? 

87.  How  many  sides  has  the  polygon  the  sum  of  whose 
^  is  4J  times  those  of  a  hexagon  ? 

88.  Find  in  inches  the  bases  of  a  trapezoid  in  which  the 
line  joining  the  middle  points  of  the  non-parallel  sides  = 
40°™  and  one  base  is  S^'™  longer  than  the  other. 

89.  How  many  sides  has  the  polygon  the  sum  of  whose 
interior  A  exceeds  the  sum  of  its  exterior  A  by  38  |^   ? 

90.  One  base  of  a  trapezoid  is  5.1™,  the  line  joining  the 
middle  points  of  the  non-parallel  sides  is  2^  times  the  other 
base  ;  find  the  other  base. 

91.  How  many  sides  has  the  polygon  each  of  whose  in- 
terior A  exceeds  its  exterior  /  by  f  |  [^  ? 

92.  How  many  sides  has  the  polygon  each  of  whose  in- 
terior ^  is  6  times  its  exterior  /  ? 

93.  Find  the  difference  in  perimeter,  in  inches,  between 
a  square  whose  side  is  1  foot  6  inches  and  a  rectangle  whose 
adjacent  sides  are  30'='"  and  60.5'=™  respectively. 


10 


GEOMETRY— NUMERICAL  PROBLEMS, 


94.  Find  the  number  of  feet  of  lime-line  of  a  tennis- 
court,  as  represented  below.  Keduce  your  answer  to 
metres.     (Log.) 


1 

*s 

a/.ff. 

^    2/.fi. 

^ 

^ 

1 

^^efi 


95.  Through  the  vertices  of  a  /\  A  B  0,  lines  are  drawn 
parallel  to  the  opposite  sides  of  the  [\  ,  thus  forming  a 
second  /\  .  Find  the  perimeter  of  the  second  l\  in  kilo- 
metres, if  the  sides  of  the  first  /\  are  5  miles,  8  miles,  and 
11  miles. 

96.  How  many  sides  has  the  polygon  each  of  whose  A 
=  162°  ? 

97.  The  perimeter  of  a  rectangle  is  8.04°*,  and  the  sides 
are  in  the  ratio  of  1  to  1-|,  find  the  lengths  of  the  sides  in 
inches. 

98.  How  many  sides  has  the  polygon  the  sum  of  whose 
interior  A  exceeds  the  sum  of  its  exterior  A  by  1080°  ? 

99.  A  man  owns  a  rectangular  garden  55"  by  34"  ;  he 
makes  a  path  3.3"  wide  around  it ;  what  is  the  perimeter 
of  the  part  that  remains  ? 

100.  Find  the  number  of  yards  of  lime-line  for  a  foot- 
ball field,  which  is  330  feet  by  160  feet,  including  all  the 
five-yard  lines.  How  long  would  it  take  a  runner  to  cover 
the  total  distance,  if  he  can  make  110  metres  in  12  seconds  ? 
(Log.) 


OEOMETRT— NUMERICAL  PROBLEMS,  11 


BOOK  II. 

1.  If  the  radii  of  two  intersecting  ®  are  3""  and  7"  re- 
spectively, what  is  the  greatest  possible  distance,  in  feet 
and  inches,  between  their  centres  ?    The  least  ? 

2.  Four  chords  are  a^-"  5°""  S*^"",  0.15  miles,  0.25'^",  and 
330  yards,  respectively.  If  one  is  a  diameter,  which  is  it  ? 
Which  of  the  others  is  nearest  to  the  centre  ?  Which 
farthest  from  it  ? 

3.  If  a  central  /  of  28°  intercepts  an  arc  of  3.2"",  find, 
in  feet  and  inches,  the  arc  intercepted  by  an  equal  /  in  an 
equal  O. 

4.  What  can  you  say  of  the  central  zi  of  a  O  which  in- 
tercept, and  the  chords  which  subtend,  two  arcs  which  are 
respectively  28  yards  and  25"  ? 

5.  In  a  given  O,  the  chord  A  B  is  5  yards  2  feet,  the 
chord  C  G  is  4.9™.  Compare  the  arcs  A  B  and  C  D,  and 
the  distances  of  the  chords  from  the  centre. 

6.  What  can  you  say  of  two  chords  whose  distances 
from  the  centre  are  13*'"  and  5  inches  respectively  ? 

7.  One  of  the  arcs  intercepted  by  two  chords,  one  of 
which  is  a  diameter,  intersecting  at  right  angles,  is  41° 
18'  4"  ;  find  the  other  arcs. 

8.  A  secant  parallel  to  a  tangent  subtends  an  arc  of 
117°  41' ;  find  the  arcs  intercepted  by  the  secant  and  the 
tangent. 

9.  One  of  the  arcs  intercepted  by  a  diameter  and  a 
parallel  secant  is  37°  30'  ;  find  the  length,  in  miles,  of  the 


12  GEOMETRY— NUMERICAL  PROBLEMS. 

arc  subtended  by  this  secant,  if  a  degree  of  the  circum- 
ference is  24^°*.     (Log.) 

10.  The  line  joining  the  centres  of  two  0,  tangent  to 
each  other  externally,  is  14"  7^"^  3*"°,  and  the  radius  of  the 
less  is  3™  8*^™  5"™,  find  the  radius  of  the  greater, 

11.  If  a  central  /  of  25°  15'  intercepts  an  arc  of  15 
feet  10  inches,  find  the  length  of  the  semi-circumference 
of  the  O.     (Log.) 

12.  How  many  degrees  in  an  /  inscribed  in  J  of  a  cir- 
cumference ? 

13.  Find  the  length  of  the  arc  intercepted  by  an  in- 
scribed /  of  20°  22^'  in  a  O  whose  circumference  is  -J-  of 
a  mile.     (Log.) 

14.  How  many  degrees  in  an  inscribed  /  which  inter- 
cepts j\  of  a  quadrant  ? 

15.  An  /  formed  by  a  tangent  and  a  chord  is  t^  |_^ ; 
how  many  degrees  in  the  intercepted  arc  ? 

16.  Find  the  length  of  the  arc  intercepted  by  a  central  / 
of  12°  15'  in  a  O  whose  circumference  =  1^"".     (Log.) 

17.  If  a  central  /  of  85°  40'  intercepts  an  arc  of  32.5'°, 
how  many  degrees  and  minutes  in  the  central  /  which  in- 
tercepts an  arc  of  65*''°  ?     (Log.) 

18.  What  part  of  a  |_^  is  an  /  between  a  tangent  and 
a  chord  intercepting  an  arc  of  JJ  of  a  semi-circumference  ? 

19.  The  /  between  two  cliords  intersecting  within  the 
circumference  is  35°,  its  intercepted  arc  is  25°  18' ;  find  the 
arc  intercepted  by  its  vertical  /_ . 


QEOMETRT— NUMERICAL  PROBLEMS. 


13 


20.  Find  the  /  between  a  secant  and  a  tangent  when 
their  intercepted  arcs  are  respectively  \  and  \  of  the  cir- 
cumference. 

21.  The  /  between  two  secants,  intersecting  without  the 
circumference,  is  58°  41',  one  of  the  intercepted  arcs  is 
230°  ;  find  the  other. 

22.  Find  the  /  between  two  tangents  when  the  inter- 
cepted arcs  are  in  the  ratio  7  :  2. 


Fig.  2. 


23.  If,  in  Fig.  2,  the  /  A  B  C  =  67°,  and  the  arc  D  0 
is  25°,  how  many  degrees  in  the  /  A  B  D  ? 

24.  In  the  same  figure,  B  G  is  a  diameter,  B  0  is  8° 
more  than  G  0  ;  find  the  /  E  B  0. 

25.  In  the  same  figure,  the  arc  D  B  is  three  and  one- 
half  times  the  arc  D  C,  and  the  /  D  B  G  =  13^°  ;  find 
the  /  D  B  C. 

26.  In  the  same  figure,  if  G  D  and  B  0  are  in  the  ratio 
3  :  7  and  the  /  D  B  0  =  15°,  how  many  degrees  in  the  / 
GBC? 

27.  In  Fig.  3,  Q  P  is  24°  less  than  a  semi-circumference, 
how  many  degrees  in  the  /  Q  M  P  ? 

28.  The  /  R  M  K  is  27°,  the  arc  R  K  is  100°  ;  how 
long  is  the  arc  T  L,  if  a  quadrant  on  this  figure  =  15"*  ? 


F  THE  \ 

ERSITY  J 


(    UNIVERSITY 


14  GEOMETRY— NUMERICAL  PROBLEMS. 

29.  The  /  R  H  K  is  70°,  the  arc  R  Q  L  is  three  times 
as  long  as  the  arc  K  P ;  find  the  number  of  degrees  in 
KP. 

30.  The  arc  R  P  T  is  10°  less  than  two-thirds  of  a  cir- 
cumference, the  /  QMTis  17°;  how  many  degrees  in 
QT? 

31.  How  many  degrees  in  the  central  /  which  inter- 
cepts an  arc  of  17*''",  when  a  quadrant  is  4^"  2*5"  5°*™  ? 

32.  The  /  between  two  tangents  from  the  same  point 
is  32°  30' ;  find  the  ratio  of  their  intercepted  arcs. 

33.  If  a  central  /  of  65°  intercepts  an  arc  of  10  feet 
5.984  inches,  how  many  metres  will  there  be  in  an  arc  of 
the  same  O  intercepted  by  a  central  /  of  211°  15'  ?   (Log.) 

34.  The  /  between  two  tangents  from  the  same  point, 
to  a  O  whose  radius  is  55^'",  is  120°  ;  how  many  inches  in 
the  chord  joining  the  points  of  tangency  ? 

35.  The  centres  of  two  ©  which  are  tangent  to  each 
other  internally  are  5  feet  8  inches  apart,  the  radius  of 
one  is  l.V  ;  find  the  radius  of  the  other. 

36.  The  chord  joining  the  points  of  tangency  of  two  in- 
tersecting tangents  forms  with  one  of  them  an  /  of  17° 
7' ;  find  the  /  between  the  tangents. 

37.  The  radii  of  two  concentric  ©  are  8  feet  2.425 
inches  and  2.25™,  respectively;  find  the  radius  of  a  0  tan- 
gent to  both.  (Two  solutions.)  Get  one  answer  in  metric 
units,  the  other  in  English  units. 

38.  The  /  between  two  chords,  one  of  which  is  a  dia- 
meter, is  ^  L^ ;  find  the  arc  subtended  by  the  less  chord. 

39.  Find  the  circumference,  in  metres,  of  a  Q  in  which 


GEOMETRY— NUMERICAL  PROBLEMS,  15 

a  central  /  of  11®  15'  intercepts  an  arc  of   3.5   inches. 
(Log.) 

40.  The  /  between  a  tangent  and  a  secant  is  8°  11', 
the  smaller  of  the  intercepted  arcs  is  56°  50'  40"  ;  find  the 
larger. 

41.  In  a  certain  O  a  central  /  of  78°  45'  intercepts  an 
arc  of  168  miles  ;  how  long  will  it  take  a  train  moving  24 
miles  per  hour  to  cover  the  circuit  ? 

42.  Two  sides  of  an  inscribed  l\  subtend  ^  and  ^  of 
the  circumference,  respectively  ;  find  the  A  of  the  /\. 

43.  One  /  of  an  inscribed  ^  is  35°,  one  of  its  sides 
subtends  an  arc  of  113°  ;  find  the  other  A  of  the  /\. 

44.  The  bases  of  a  trapezoid  subtend  arcs  of  100°  and 
140°,  respectively  ;  find  its  A  and  the  /  made  by  the  non- 
parallel  sides  produced. 

45.  How  long  would  it  take  a  train  running  40  miles  an 
hour  to  go  round  a  O  in  which  a  central  /  of  15°  inter- 
cepts an  arc  of  7.2^"  ?     (Log.) 

46.  The  numbers  of  degrees  in  the  arcs  subtended  by 
the  sides  of  a  pentagon,  in  order,  are  consecutive  ;  find  the 
A  of  the  pentagon. 

47.  The  arcs  subtended  by  three  consecutive  sides  of  a 
quadrilateral  are  87°,  95°,  115°  ;  find  the  A  of  the  quadri- 
lateral ;  the  A  made  by  the  intersection  of  the  diagonals  ; 
and  the  A  made  by  the  opposite  sides  of  the  quadrilateral, 
when  produced. 

48.  Find  the  /  made  by  the  radii  and  the  line  joining 
the  points  of  contact  of  two  tangents  drawn  through  a 


16  OEOMETRT— NUMERICAL  PROBLEMS. 

point  6  inches  from  the  circumference  of  a  O  of  6-inch 
radius. 

49.  Find  the  A  of  an  isosceles  /\,  if  the  arc  subtended 
by  one  of  the  equal  sides  is  33°  more  than  1.6  times  the 
arc  subtended  by  the  base. 

50.  An  /  formed  by  a  diagonal  and  a  base  of  an  in- 
scribed trapezoid  is  20°  30' ;  find  the  A  made  by  the  in» 
tersection  of  the  diagonals. 

51.  Over  how  many  degrees  of  arc  of  a  O  whose  circum- 
ference is  435'^"'  will  a  train,  moving  60  miles  per  hour,  go 
in  15  minutes  5  seconds  ?    (Log.) 

52.  Three  consecutive  A  of  an  inscribed  quadrilateral 
are  140°  30',  80°  30',  and  29°  30' ;  find  the  numbers  of  de- 
grees in  the  arcs  subtended  by  the  four  sides. 

53.  If  it  takes  light  8  minutes  to  come  from  the  sun 
to  the  earth,  which  distance  is  the  same  as  57.3°  of  the 
earth^s  orbit,  how  long  would  it  take  it  to  go  the  length 
of  the  entire  orbit,  supposing  the  orbit  a  O  ?     (Log.) 

54.  Three  consecutive  ^  of  a  circumscribed  quadri- 
lateral are  85°,  122°,  111°  ;  find  the  number  of  degrees  in 
each  /  of  the  inscribed  quadrilateral  made  by  joining  {he 
points  of  contact  of  the  sides  of  the  circumscribed  quadri- 
lateral. 

55.  Find  the  circumference  of  a  O  in  which  a  train  go- 
ing 60  miles  an  hour  goes  over  an  arc  of  1°  35'  in  17 
seconds.     (Log.) 

56.  Two  arcs  subtended  by  two  adjacent  sides  of  an  in- 
scribed quadrilateral  are  127°  and  68°  30',  and  the  /  be- 
tween the  diagonals,  which  intercepts  the  arc  of  68°  30',  is 
77°  30' ;  find  the  A  of  the  quadrilateral. 


GEOMETRY— NUMERICAL  PROBLEMS. 


17 


57.  If  a  star  makes  a  complete  circuit  of  the  heavens  in 
23  hours  56  minutes,  through  what  arc  will  it  go  between 
9.12  P.M.  and  12.13  a.m.  ?     (Log.) 

58.  If  the  earth  in  revolving  about  the  sun  moves  65,500 
miles  per  hour  in  its  orbit,  find  the  entire  length  of  this 
orbit,  remembering  that  it  takes  365  days  6  hours  9 
minutes  9  seconds  to  make  a  complete  revolution.    (Log.) 

59.  If  Jupiter  is  476,000,000  miles  from  the  sun,  and 
the  length  of  its  orbit  is  three  and  one-seventh  times  the 
diameter  of  its  orbit,  and  its  period  of  revolution  is  11 
years,  315  days,  what  is  its  hourly  motion  in  its  orbit  ? 
(Log.) 

60.  If  the  earth's  radius,  3,963  miles,  is  equal  to  the 
length  of  an  arc  of  57'  of  the  moon's  orbit  about  the  earth, 
what  is  the  distance  to  the  moon,  considering  the  orbit  a 
O  and  the  circumference  three  and  one-seventh  times  the 
diameter  ?    (Log.) 


BOOK   III. 

1.  In  Fig.  4,  B  0  =  52™,  A  0  =  28'",  A'  B'  is  ||  to  A  B, 
C  B'  =r  IS'"  ;  find  C  A'  and  A'  A. 

2.  If,  in  the  same  figure,  0  A'  =  10  feet.  A'  A  =  12  feet 
4  inches,  and  B'  C  =  16  feet  3  inches,  what  is  the  length 
of  OB? 


Fig.  4. 


Fig.  6. 


18  GEOMETRY— NUMERICAL  PROBLEMS. 

3.  In  Fig.  5,  A  B  =  18.7™,  B  C  =  29.4™,  A  C  =:  40.4™, 
and  B  D  is  the  bisector  of  the  /ABC;  find  A  D  and 
DC.     (Log.) 

4.  If,  in  Fig.  5,  A  D  =  3  feet  5  inches,  A  B  =  4  feet  2 
inches,  and  B  C  =  7  feet,  find  the  length  of  A  C. 

5.  In  Fig.  6,  C  D  is  the  bisector  of  the  /  A  C  F, 
B  E  =  3.3'*'",  A  C  =  e*''",  B  C  =  4.1'*">  ;  find  A  B  in  yards. 
(Log.) 

G.  If,  in  Fig.  6,  A  C  =  65  yards,  A  B  =  48  yards,  B  C 
=  35  yards ;  find  B  E  in  metres.     (Log.) 

7.  If,  in  Fig.  6,  A  E  =  18  feet  6  inches,  B  C  =  14  feet, 
and  B  E  =  14  feet  2  inches ;  find  in  metres  the  lengths 
of  A  C  and  A  B.     (Log.) 


8.  The  sides  of  a  A  are  a^  =  15",  h=  12",  c  =  10"  ;  find 
the  segments  into  which  each  side  is  divided  by  the  bisec- 
tor of  the  opposite  / . 

9.  Find  the  segments  into  which  each  side  is  divided 
by  the  bisector  of  an  exterior  /  in  the  preceding  problem. 

10.  The  homologous  sides  of  two  similar  A  are  5 
feet  3  inches  and  4  feet  5  inches,  respectively.  If  the  al- 
titude to  the  given  side  of  the  first  is  3  feet  9  inches,  find 
the  homologous  altitude  in  the  second. 

11.  The  sides  of  a  A  are  4"  6^™,  6"  1^'",  and  8"  ;  the 
homologous  sides  of  a  similar  A  are  a,  305'",  c ;  find  a 
and  c. 

12.  In  the  A  A  B  C  and  A'  B'  C,  A  =  59°  =  A',  J  =  3 
feet  6  inches,  c  =  13  feet,  J' =  5.6",  c' =  20.8".  Show 
what  relation,  if  any,  these  A  bear  to  each  other. 


QEOMETRT— NUMERICAL  PROBLEMS.  19 

13.  The  perimeters  of  two  similar  polygons  are  88™ 
and  396™,  respectively.  One  side  of  the  first  is  15  yards 
4  feet  2.4  inches;  find  the  homologous  side  of  the  sec- 
ond.    (Log.) 

14.  The  sides  of  two  A  are,  respectively,  4*^,  9^™,  11^", 
and  1.2  miles,  2.7  miles,  3.3  miles.  Show  by  your  work 
any  relation  which  may  exist  between  these  A. 

15.  One  of  the  altitudes  of  a  /\  =  1.5™  ;  find  the 
homologous  altitude  of  a  similar  ^,  if  the  perimeters  of 
the  two  A  are  respectively  15  feet  and  24  feet. 

16.  A  series  of  straight  lines  passing  through  the  point 
0  intercept  segments,  on  one  of  two  parallel  lines,  of  15 
feet,  18  feet,  24  feet,  and  32  feet,  the  segment  of  the  other 
parallel,  corresponding  to  24  feet,  is  16  feet ;  find  the  other 
segments. 

17.  Two  homologous  sides  of  two  similar  polygons  are 
35™  and  50™,  respectively.  The  perimeter  of  the  second  is 
8"™.     What  is  the  perimeter  of  the  first  ? 

18.  The  legs  of  a  right  /\  are  3™  and  4™  ;  find,  in 
inches,  the  difference  between  the  hypotenuse  and  the 
greater  leg.  Find  also  the  segments  of  the  hypotenuse 
made  by  the  perpendicular  from  the  vertex  of  the  right 
/  ;  and  this  perpendicular  itself. 

19.  In  a  O  whose  diameter  is  16™,  find  the  length  of  the 
chord  which  is  4™  from  the  centre. 

20.  The  sides  of  a  A  are  30"=™,  40<=™,  and  45*=™  ;  find  the 
projection  of  the  shortest  side  upon  the  longest. 

21.  Is  the  l\  of  20  acute,  right,  or  obtuse  ?  Which 
would  it  be  if  the  sides  were  30*=™,  40*=™,  55'='"  ?    Find  the 


OEOMETBT—NUMEMIGAL  PROBLEMS. 


projection  of  the  shortest  side  upon  the  medium  side  in 
the  latter  /\. 

22.  A  tangent  to  a  O  whose  radius  is  1  foot  6  inches,  from 
a  given  point  without  the  circumference,  is  2  feet ;  find  the 
distance  from  the  point  to  the  centre. 

23.  In  the  A  A  B  C,  a  =  14^  h  =  17™,  c  =  22™  ;  is 
the  /  C  acute,  right,  or  obtuse  ? 

24.  To  find  the  altitude  oidi,  /\m  terms  of  its  sides. 


(1)  ^2  _  ^2  _  g  j)2^  (The  square  of  either  leg  of  a  right 
A  is  equal  to  the  square  of  the  hypotenuse  minus  the 
square  of  the  other  leg.) 


&2  =  «2  +  c2  -  2a  X  B  D 


Solving  for  B  D,  B  D 


The  square  of  the  side  opposite 
the  acute  /  of  a  A  is  equal  to 
the  sum  of  the  squares  of  the 
other  two  sides  minus  twice  one 
of  them  by  the  projection  of 
the  other  upon  it. 

a2  +  g2  -  h" 
2a 


GEOMETRY— NUMERICAL  PROBLEMS.  21 

Substituting  in  (1), 

~\  2a  )  \  2a  ) 

Ha  +  cf-bH   rb^-{a-cf-i 
~L        2a       i   I      2a        J 
_(a  +  g  +  Z>)  ja  +  c—b)       {b  +  a-c)  (b-a  +  c) 
~  2a  ^  2a  ' 

Let  2s  =  a  +  b-\-c, 
Subtracting  2c=2cy 
2s—2c=2{s—c)=a  +  b—c. 
Similarly,  2{s—a)=b-\-c—a, 
and2{s—b)=a-\-c—b. 
Substituting  we  have 

2^  X  2(s—b)       2(s-c)  X  2{s—a)    4.s{s-a)  (s-b)  (s-c). 

"'-        2a  ^  2a  "  a' 

2      . 

Extracting  the  square  root,  h=-  V  s{s—a)  {s—b){s—c), 

a 

2     / 

Similarly,  ^'''—h    ys{s—a)(s—b)(s—c), 

and  //'=?  V  s(s-a)(s-b){s-c), 

c 

li'  and  li"  representing  the  altitude  of  the  /\  upon  b  and 
c,  respectively. 

25.  To  find  the  radius  of  the  circumscribed  O  in  terms 
of  the  sides  of  the  [\. 


Fig.  8. 


22  GEOMETRY— NUMEBIGAL  PROBLEMS. 

«c=2RxBD.     (Fig.  8.) 

(The  product  of  two  sides  of  a  /\  is  equal  to  the  dia- 
meter of  the  circumscribed  O  multiplied  by  the  altitude  to 
the  third  side.) 

But  by  24,  B  D:=-  V  s(s-a)  (s-b)  (s-c) 


b 
b 


4J{      

Hence  ac=  —  Vs(s-a)  (s-b)  (s-c)' 


T.  a  b  c 

and  K  = 


4  y\/  s(s—a)  {s—b)  (s—c) 

26.  To  find  the  bisectors  of  the  A  ot  a,  l\m  terms  of  the 

sides. 

(1)  a  c=x^+A  D  X  D  C.     (Fig.  9.) 

(The  product  of  two  sides  of  a  /\  is  equal  to  the  square 
of  the  bisector  of  the  included  / ,  plus  the  product  of  the 
segments  of  the  third  side  made  by  the  bisector.) 

Transposing  in  (1),  (2)  x^=a  c—ABxB  C. 

Tj   ,  CD     a 

But  rTT^-- 

DA    c 

(The  bisector  of  an  /  of  a  /\  divides  the  opposite  side 
into  segments  proportional  to  the  adjacent  sides.) 

T5                ...       D  C  +  A  D    a  +  c 
By  composition  — ^— — —  = , 

and  5^-.=^ — = or 

AD  c 

_L=^Lh£.,  and -l_=^±f. 
J)  C       a    '        AD       c 

Whence  D  C=— ,  and  A  J)=—. 

a+c  a+c 


OEOMETRT— NUMERICAL  PROBLEMS.  23 

Substituting  in  (2)  we  have 

_  a  c{c+a-\-h)  (c+a—h) 
{a+cf 
(Substituting  as  in  24.) 

_«cx25x2(s— Z»). 

Extracting  the  square  root. 

Similarly,  o(fz=ij-—  V  h  c  s  (s—a)' 

_2_ 
a  +  6 


and  ic"=-^  V abs(s- 


Note. — In  a  right  /\  (hypotenuse  c  and  legs  a,  h)  the 
formula  a=  "s/  c^—h^  and  h—  ^/  c^—c?y  should  be  written 
a='\/(c  +  ^)  {c—h),  and  1= '\/ {c-{-a){c—d),  when  loga- 
rithms are  to  be  employed. 

27.  The  chord  A  B,  which  is  4.2™  long,  divides  the  chord 
0  D  into  segments  which  are  1.4"  and  2.1™,  respectively. 
Find  the  segments  of  A  B  made  by  C  D. 

28.  The  sides  of  a  /\  are  25  yards,  30  yards,  35  yards. 
Find  the  length  of  the  median  *  to  the  side  of  30  yards, 
and  its  projection  upon  the  same. 

29.  Find  the  diameter  of  the  O  circumscribed  about 
the  l\  two  of  whose  sides  are  3  feet  4  inches  and  4  feet  6 
inches,  and  the  perpendicular  to  the  third  side  from  the 
opposite  vertex  is  2  feet  3  inches. 

30.  Find  the  length  of  the  bisector  of  the  opposite  /  to 
the  least  side  in  the  A  whose  sides  are  24™,  20'™,  11'™ ;  the 

*  A  median  is  a  line  from  a  vertex  o-f  a  A  to  the  middle  point  of  the 
opposite  side. 


24  GEOMETRY— NUMERICAL  PROBLEMS. 

three  altitudes  of  the  /\  ;  and  the  radius  of  the  circum- 
scribed O.     (Log.) 

31.  Two  secants  from  the  same  point  without  a  0  are 
25*='"  and  35'='°.  If  the  external  segment  of  the  less  is  7'^ 
find  the  external  segment  of  the  greater. 

32.  A  secant  from  a  given  point  without  a  O  and  its 
external  segment  are  2  feet  4  inches  and  7  inches^  respec- 
tively ;  find  the  length  of  the  tangent  to  the  O  from  the 
same  point. 

33.  The  greatest  distance  of  a  chord  of  11  feet  from  its 
arc  is  6  inches  ;  find  the  diameter  of  the  O. 

34.  Two  sides  of  a  ^,  inscribed  in  a  0  whose  radius  is 
15  inches,  are  9  inches  and  25  inches  ;  find  the  perpen- 
dicular to  the  third  side  from  the  opposite  vertex. 

35.  Find  the  greater  segments  of  a  line  of  36""  when  it 
is  divided  internally  and  externally  in  extreme  and  mean 
ratio. 

36.  Find  a  mean  proportional  to  two  lines  which  are 
5*^"  and  2"°  long,  respectively. 

37.  Find  a  fourth  proportional  to  the  lines  a,  h,  c,  when 
^=65'=^  J=42^^  c=26*='°. 

38.  Find  a  third  proportional  to  m  and  n,  when  m=:17^"" 
and  ^=51^"". 

39.  The  chords  A  B  and  C  D  intersect  at  E  ;  A  E  = 
15"^",  B  E=46'''",  C  0  =  115'^'"  ;  find  C  E  and  D  E. 

40.  Find  the  distance  from  a  given  point  to  the  circum- 
ference of  a  0  whose  radius  is  9  inches,  if  the  tangent  to 
the  0  from  the  given  point =1  foot. 

41.  If,  in  the  preceding  problem,  another  tangent  were 


GEOMETRY— NUMERIOAL  PROBLEMS,  25 

drawn  from  the  same  point,  what  would  be  the  length  of 
the  line  joining  the  points  of  contact  of  these  two  tan- 
gents ? 

42.  The  segments  of  a  transversal  made  by  lines  passing 
through  a  common  point  are  1  foot  3  inches,  1  foot  9  inches, 
and  2  feet  11  inches,  respectively.  If  the  least  segment  of 
a  parallel  to  this  transversal,  intercepted  by  the  same  lines, 
is  30*="",  find  the  other  segments. 

43.  If  a  gate-post  5  feet  high  casts  a  shadow  17  feet  long, 
how  high  is  a  house  which,  at  the  same  time,  casts  a  shadow 
221  feet  long  ? 

44.  A  baseball  diamond  is  a  square  with  90  feet  to  a 
side  ;  find  the  distance  across  from  first  base  to  third. 

45.  The  projections  of  the  legs  of  a  right  l\  upon  the 
hypotenuse  are  8'""  and  9'^'"  ;  find  the  shorter  leg. 

46.  In  a  O  whose  radius  is  41  feet  are  two  parallel 
chords,  one  80  feet,  the  other  18  feet.  Find  how  far  apart 
these  two  chords  are.     (Two  solutions.) 

47.  If  a  chord  of  75''"  subtends  an  arc  of  m°  in  a  O 
whose  radius  is  415'='",  how  long  a  chord  will  subtend  an 
arc  of  m°  in  a  O  whose  radius  is  33.20""  ?    (Log.) 

48.  The  sides  of  a  A  are  1,789"^,  4,231^  and  3,438"^ ; 
find  the  three  altitudes  and  the  diameter  of  the  circum- 
scribed O.     (Log.) 

49.  The  altitude  of  an  equilateral  /\  is  45  feet,  what  is 
the  length  of  a  side  in  feet  and  inches  ? 

50.  Find  the  radius  of  the  O  in  which  a  chord  of  40.5'" 
is  14.4""  from  the  centre.  Find  also  the  distances  from 
one  end  of  this  chord  to  the  ends  of  the  diameter  perpen- 
dicular to  it. 


26  GEOMETRY— NUMERICAL  PROBLEMS. 

51.  The  greater  segments  of  a  line  divided  internally  in 
extreme  and  mean  ratio  is  1  foot  6  inches  ;  find  the  length 
of  the  line. 

52.  The  projections  of  the  legs  of  a  right  /\  upon  the 
hypotenuse  are  27'''"  and  48'^"' ;  find  the  lengths  of  the  legs. 

53.  Find  the  width  of  a  street,  where  a  ladder  95.8  feet 
long  will  reach  from  a  certain  point  in  the  street  to  a  win- 
dow 67.3  feet  high  on  one  side,  and  to  one  82.5  feet  high 
on  the  other  side.     (Log.) 

54.  Find  the  diameter  of  a  O  in  which  the  chord  of 
half  the  arc  subtended  by  a  chord  of  dO'"^  is  17*=". 

55.  Find  the  altitude  of  an  equilateral  /\  whose  side= 

2.2"". 

56.  What  is  the  diameter  of  a  O  when  the  point  from 
which  a  tangent  of  6  feet  is  drawn  is  8  inches  from  the 
circumference  ? 

57.  The  sides  of  a  A  are  185",  227'",  and  242""  ;  find  the 
three  altitudes,  the  bisectors  of  the  three  A,  and  the 
radius  of  the  circumscribed  O.     (Log.) 

58.  The  sides  of  a  trapezoid  are  437.3  feet,  91  feet,  291. 7 
feet,  and  91  feet ;  find  the  altitude  of  the  trapezoid  and 
the  diagonals. 

59.  The  sides  of  a  parallelogram  are  24 J  miles  and  31 J 
miles,  and  one  of  the  diagonals  is  28  miles ;  find  the  num- 
ber of  kilometres  in  the  other  diagonal. 

60.  If  a  chord  of  2  feet  is  5  inches  from  the  centre  of  a 
O,  what  is  the  distance  of  a  chord  whose  length  is  10 
inches  ? 

61.  One  side  of  a  A  is  136"'°,  the  altitude  of  the  A  to 
the  second  side  is  102'^'",  the  diameter  of  the  circumscribed 
O  is  184'='" ;  find  the  third  side  of  the  £\. 


OEOMETRT— NUMERICAL  PROBLEMS.  27 

62.  The  common  chord  of  two  intersecting  ©  whose 
radii  are  2  feet  1  inch  and  1  foot  9  inches  is  1  foot  2 
inches  ;  find  the  distance  between  their  centres. 

63.  Is  the  A  whose  sides  are  38"°,  36™,  12^  acute,  right, 
or  obtuse  ? 

64.  In  the  /\  whose  sides  are  11"",  13"*,  14",  find  the 
segments  into  which  the  side  14  is  divided  by  the  perpen- 
dicular from  the  opposite  vertex. 

65.  Find  the  legs  of  a  right  A  when  their  projections 
upon  the  hypotenuse  are  11.16  feet  and  19.84  feet. 

QQ.  The  sides  of  a  A  are  23  feet,  27  feet,  38  feet ;  find 
the  length  of  the  median  to  the  longest  side  and  its  projec- 
tion upon  the  longest  side. 

67.  What  is  the  longest  and  shortest  chord  that  can  be 
drawn  through  a  point  15"°  from  the  centre  of  a  ©  whose 
radius  is  39'="  ? 

68.  How  long  is  the  shadow  of  a  house  23»"  high,  when 
a  stake  4  feet  high  casts  a  shadow  2  feet  6  inches  long  ? 
(Log.) 

69.  Find  the  length  of  the  common  tangent  of  two  © 
which  cuts  the  line  joining  their  centres,  when  this  line  is 
2  feet  and  the  radii  of  the  ©  are  5  inches  and  3  inches. 

70.  The  greater  leg  of  a  right  A  ^^  1  inch,  and  the  dif- 
ference between  the  hypotenuse  and  the  less  leg  is  J 
inch  ;  find  the  hypotenuse,  the  less  leg,  the  perpendicular 
from  the  vertex  of  the  right  /  to  the  hypotenuse,  and  the 
segments  of  the  hypotenuse  made  by  this  perpendicular. 

71.  Find  the  product  of  the  segments  of  any  chord 
passing  through  a  point  8"  from  the  centre  of  a  0  whose 
diameter  is  20"°, 


28  QEOMETRT— NUMERICAL  PROBLEMS, 

72.  Through  a  point  21'='"  from  the  circumference  of  a  O 
is  drawn  a  secant  84'"  long.  The  chord  part  of  this 
secant  is  SI""*.     Find  the  radius  of  the  O. 

73.  The  diagonals  A  C  and  B  D  of  an  inscribed  quadri- 
lateral intersect  at  E,  A  0  is  59^  B  E  35^",  and  D  E  IS"* ; 
find  A  E  and  C  E. 

74.  What  is  the  length  of  a  tangent  drawn  from  a  point 
4  inches  from  the  circumference  of  a  O  whose  radius  is  3 
feet  9  inches  ? 

75.  Find  the  diameter  of  a  O  in  which  two  chords,  30 
feet  and  40  feet  long,  parallel  and  on  opposite  sides  of  the 
diameter,  are  35  feet  apart. 

76.  The  smaller  segment  of  a  line  divided  externally  in 
extreme  and  mean  ratio  is  12'""  ;  find  the  length  of  the 
greater  segment. 

77.  Two  sides  of  a  ^  are  16^"  and  O^"",  and  the  median 
to  the  first  side  is  11^"" ;  find  the  length  of  the  third  side 
in  miles. 

78.  In  the  preceding  problem,  find  the  lengths  of  the 
projections  of  the  median  and  the  second  and  third  side 
upon  the  first  side. 

79.  Find  the  lengths  of  the  projections  of  each  side 
upon  the  other  two  sides  in  a  /\  whose  sides  are  6"",  8"",  and 
12™. 

80.  How  far  apart  are  two  parallel  chords  48  feet  and  14 
feet  long  in  a  O  whose  diameter  is  50  feet,  if  they  are  on 
the  same  side  of  the  centre  ? 


OEOMETBY— NUMERICAL  PROBLEMS,  29 


BOOK  IV. 

1.  Find  the  area  of  a  rectangle  whose  base  and  altitude 
are  37  feet  and  14  feet. 

2.  What  is  the  area  of  a  parallelogram  whose  base  and 
altitude  are  13™  and  18™  ? 

3.  How  many  hektares  in  a  rectangular  field  53^""  by 
29Dm  p 

4.  Find  the  width  of  a  rectangular  field  containing  an 
acre,  if  the  length  is  176  yards. 

5.  How  many  acres  in  a  parallelogram  whose  base  and 
altitude  are  17«"  and  13»"  ?    (Log.) 

6.  How  many  rods  in  the  side  of  a  square  field  contain- 
ing a  hektare  ?    (Log.) 

7.  How  many  metres  in  the  side  of  a  square  field  con- 
taining an  acre  ?    (Log.) 

8.  A  rectangle  which  is  7  times  as  long  as  it  is  wide  con- 
tains 32  square  rods  ;  find  its  width  and  length. 

9.  Find  the  area  of  the  surface  of  a  flower-bed  4.55™ 
long  and  2.75™  wide. 

10.  The  perimeter  of  a  rectangle  is  24™,  and  the  length 
is  9.2™  ;  find  the  breadth  and  the  area  of  the  rectangle. 

11.  What  is  the  ratio  of  the  areas  of  two  rectangular 
fields,  one  of  which  is  231™  long  and  87™  wide,  and  the 
other  58™  wide  and  110™  long  ? 

12.  Two  rectangles  have  the  same  altitude,  and  the  area 
of  the  first  is  62  acres  and  the  area  of  the  second  38  acres. 


30  GEOMETRY— NUMERICAL  PROBLEMS. 

If  the  base  of  the  first  is  570  rods,  what  is  the  base  of  the 
second  ? 

13.  What  part  of  a  mile  is  the  perimeter  of  a  square 
hektare  (1^""  =  !•""«)  ? 

14.  The  perimeter  of  a  rectangle  is  6  feet,  and  the  length 
is  3  times  the  breadth  ;  find  the  length,  the  breadth^  and 
the  area  of  the  rectangle. 

15.  If  the  perimeter  of  a  rectangle  is  26™  and  its  length 
is  2.5°"  more  than  its  breadth,  find  its  length,  breadth,  and 
area. 

16.  A  parallelogram  whose  area  is  one  acre  has  a  base  of 
60  rods  ;  and  one  whose  area  is  1"*  has  the  same  altitude  ; 
find  the  base  of  the  latter.     (Log.) 

17.  Find  the  side  of  a  square  equivalent  in  area  to  a 
rectangle  whose  base  is  4  feet  6  inches  and  whose  altitude 
is  6  inches. 

18.  What  is  the  altitude  of  a  rectangle  whose  base  is 
23",  equivalent  to  a  square  whose  area  is  5.06*  ? 

19.  Find  the  area  of  a  /\  whose  base  and  altitude  are 
respectively  3  feet  2  inches  and  5  yards  1  inch. 

20.  Find  the  base  and  altitude  of  a  rectangle  whose  pe- 
rimeter is  54™  and  whose  area  is  182'^"'. 

21.  Find  the  side  of  a  square  whose  area  is  18  square 
yards  7  square  feet. 

22.  Find  the  area,  in  acres,  of  a  rectangle  whose  pe- 
rimeter is  156°"  and  whose  dimensions  are  to  each  other 
as  6  :  7.     (Log.) 

23.  A  l\  whose  base  is  35'='"  contains  .525'"  ;  how  many 
square  inches  in  a  /\  whose  homologous  base  is  14'^"  ? 
(Log.) 


GEOMETRY— NUMERICAL  PROBLEMS.  31 

24.  Find  the  area  of  an  equilateral  l\  whose  side  is  8 
feet. 

25.  Find  the  difference  in  area  between  a  /\  whose  base 
and  altitude  are  each  1  yard,  and  a  /\  whose  sides  are 
'each  1"".     (Log.) 

26.  The  bases  of  a  trapezoid  are  7.32""  and  8.45"",  and 
the  altitude  is  4.4"  ;  find  the  area  in  ares. 

27.  The  altitude  of  an  equilateral  A  is  6  feet  3  inches  ; 
find  the  area. 

28.  To  find  the  area  of  a  A  i^  terms  of  its  sides. 

J 


Let  K  =  the  area  of  the  /\- 
(1)   K  =\  ah. 

(The  area  of  a  A  is  equal  to  one-half  the  product  of  its 
base  and  altitude.) 

By  24,  Book  III.,     h  =  -  Vsis-a)  (s-b)  (s-c)  ' 

Substituting  in  (1),     K  =^^x-  Vs  (s~a)  (s—b)  {s—c)\  or, 

K=  V  8(8—0)  {s—b)  (s—c)' 

29.  To  find  the  area  of  a  A  i^  terms  of  its  sides  and 
the  radius  of  the  circumscribed  0 . 

By  25,  Book  IIL,       R  =  ,    .  ,         ""^f    ,,  ,         « 
^  Ws{s-a)  (s-b)  (s-c) 


32  GEOMETRY— IfUMERICAL  PROBLEMS. 

Substituting  K  for  its  value  as  found  in  the  preceding 
article, 

abc 
Solving,  ^  =  VB' 

30.  To  find  the  area  of  a  /\  in  terms  of  its  sides  and 
the  radius  of  the  inscribed  O. 


Fig.  11. 

By  drawing  lines  from  the  centre  of  the  O  to  the  ver- 
tices we  form  three  A  whose  common  vertex  is  0,  whose 
bases  are  a,  i,  c,  the  sides  of  the  given  /\,  and  whose  alti- 
tudes are  each  r,  the  radius  of  the  inscribed  O. 
Now,  Area  AOC  =  i  br, 

''     AOB  =  i  cr, 

"     BOO  =  i  ar. 
Adding,  ''    ABO  =  ^{a+b  +  c)r. 

Substituting  K  for  area  ABO,  and  s  for  ^  («  +  ^  4-  c), 

K=  rs. 

From  this  equation,  r  =  —  ;  i.e.,  the  radius  of  the  in- 

s 

scribed  O  equals  the  area  of  the  [\  divided  by  one-half  the 

perimeter. 

*  Hereafter  this  form  of  the  formula  for  R  should  be  held  in  mind. 


GEOMETRY— NUMERICAL  PROBLEMS. 


33 


31.  To  find  the  area  of  a  /\  in  terms  of  its  sides  and  the 
radius  of  an  escribed  ©.* 

(1)  Area  AOB  =  \  cr\ 

(2)  ''     AOO  =  i  br\ 

(3)  "     BOO  =  i  ar\ 
Subtracting  (3)  from  the  sum  of 
(1)  and  (2), 

Area  ABO  =  ^  {b  ■{-  c  —  a)r';  or, 
K  =  (s-  a)r'. 
Similarly,  K=  (5 -^>)/', 
K  =  (s-  cy, 
/'  and  r'"  representing  the  radii 
of  the  escribed  ®   tangent  to  h 
and  c,  respectively. 

zr 

These  three  formulae  give  r'= > 

s     a 


Fig.  12. 


K 


,  from  which  the  radii  of  the  escribed 


0  can  be  found  when  the  sides  of  a  /\  are  given. 

32.  The  sides  of  a  A  are  :  a  =  21",  I  =  17",  c  =  10"  ; 
find  the  area  of  the  /\  and  the  radii  of  the  circumscribed, 
inscribed,  and  escribed  ®. 

33.  Find  the  difference  in  area  between  a  rectangle  4 
times  as  long  as  wide,  with  a  perimeter  of  100  yards  and  a 
square  whose  perimeter  is  80  yards. 

34.  A  man  has  a  rectangular  piece  of  ground  55"  by 
110".  After  a  path  4.5"  wide  is  made  around  it,  is  the 
part  left  more  or  less  than  an  acre  ?    How  much  ? 

35.  The  bases  of  a  trapezoid  are  13.2'"  and  15.6'",  and 
the  altitude  is  1  yard  2  inches  ;  find  the  area  in  centares. 

♦  An  escribed  O  is  a  O  tangent  to  one  side  of  a  A  and  the  prolonga- 
tions of  the  other  two  sides. 


84  GEOMETRY— NUMERICAL  PROBLEMS. 

36.  The  side  of  a  square  is  2  feet ;  find  the  sides  of  an 
equivalent  rectangle  whose  base  is  4  times  its  altitude. 

37.  The  area  of  a  A  is  ll^*^"",  its  base  is  14"  ;  find  the 
area  of  a  similar  /\  whose  homologous  base  is  8°*. 

38.  Find  the  dimensions  of  a  rectangle  whose  perimeter 
is  8  feet  4  inches,  and  whose  area  is  4  square  feet  13  square 
inches. 

39.  Through  the  middle  of  a  rectangular  garden,  156"°  by 
140"",  run  two  paths  at  right  angles  to  each  other  and 
parallel  to  the  sides,  the  longer  one  0.8"  wide,  the  shorter 
1.2"  wide  ;  find  the  area  not  taken  up  by  the  paths. 

40.  The  sides  of  a  A  are  :  «  =  588  feet,  h  —  708  feet, 
c  =  294  feet ;  find  the  area  of  the  /\  and  the  radii  of  the 
circumscribed,  inscribed,  and  escribed  0.     (Log.) 

41.  The  area  of  a  rhombus  is  360<^%  one  diagonal  is 
7.2"^"  ;  find  the  other. 

42.  The  area  of  a  polygon  is  5f  times  the  area  of  a 
similar  polygon.  If  the  longest  side  of  the  larger  polygon 
is  40",  what  is  the  longest  side  of  the  smaller  polygon  ? 

43.  Find  the  area  of  a  square  whose  diagonal  is  30  feet. 

44.  Find  the  number  of  square  feet  in  an  equilateral  /\ 
whose  side  is  one  metre.     (Log.) 

45.  Find  the  side  in  kilometres,  of  an  equilateral  ^ 
whose  area  is  47  acres.     (Log.) 

46.  Find  the  side  of  a  square  equivalent  to  the  differ- 
ence of  two  squares  whose  sides  are  115"  and  69". 

47.  Find  the  area,  in  square  feet,  of  an  isosceles  right  ^ 
if  the  hypotenuse  is  25".     (Log.) 


GEOMETRY— NUMERICAL  PROBLEMS.  35 

48.  The  sides  of  a  ^  are  10  feet,  17  feet,  and  21  feet. 
Find  the  areas  of  the  two  parts  into  which  the  l\  is 
divided  by  the  bisector  of  the  /  formed  by  the  first  two 

sides. 

49.  The  side  of  a  rhombus  is  39"*,  and  its  area  is  540ca  ; 
find  its  diagonals. 

50.  The  area  of  a  trapezoid  is  13  acres,  and  the  sum  of 
its  bases  is  813  yards ;  find  its  altitude. 

51.  Find  the  area,  in  acres,  of  a  right  l\  whose  hy- 
potenuse is  36"""  and  one  leg  28.8"".     (Log.) 

52.  Find  the  ratio  of  the  areas  of  two  A  which  have  a 
common  /,  when  the  sides  including  this  /  in  the  first 
are  131""  and  147",  and  in  the  second  are  211  feet  and  287 
feet.     (Log. ) 

53.  Two  homologous  sides  of  two  similar  polygons  are 
21"°"  and  35""" ;  the  area  of  the  greater  polygon  is  525"* ; 
what  is  the  area  of  the  smaller  polygon  ? 

54.  Find  the  area  of  a  quadrilateral  whose  sides  are  8", 
10™,  12*",  6°",  and  one  of  whose  diagonals  is  14™. 

55.  On  a  map  whose  scale  is  1  inch  to  a  mile,  how  many 
hektares  would  be  represented  by  a  square  centimetre  ? 
(Log.) 

56.  The  homologous  altitudes  of  two  similar  A  are  9™ 
and  21™,  and  the  area  of  the  smaller  is  405  square  feet ; 
find  the  area  of  the  larger. 

57.  The  area  of  a  trapezoid  is  84"*,  its  altitude  3.5"™, 
and  one  base  20"™  ;  find  the  other  base. 


36  GEOMETRY— NUMERICAL  PROBLEMS. 

58.  The  areas  of  two  A  are  144  square  yards  and  108 
square  yards.  Two  sides  of  the  second  are  12  yards  and  21 
yards,  and  one  side  of  the  first  is  9  yards.  Find  a  second 
side  of  the  first,  which,  with  the  side  9  yards  includes  an 
/  equal  to  the  /  of  the  second  included  by  the  sides  12 
yards  and  21  yards. 

59.  Find  the  area  of  a  square  whose  diagonal  is  8". 

60.  Find  the  difference  in  perimeter  between  a  rectangle 
whose  base  is  16  feet  and  an  equivalent  square  whose  side 
is  12  feet. 


61.  Find  the  diagonals  of  a  rhombus  whose  side  is  6  feet 
1  inch  and  whose  area  is  9  square  feet  24  square  inches. 

62.  Find  the  area  of  a  trapezoid  whose  parallel  sides 
are  28™  and  SS"",  and  whose  non-parallel  sides  are  12"" 
and  13". 

63.  Find  the  dimensions  of  a  rectangle  whose  area  is 
1,452  square  feet  and  one  of  whose  sides  is  |  its  diagonal. 

64.  The  sides  of  a  A  are  26"",  28",  30"" ;  find  its  area, 
the  three  altitudes,  and  the  radii  of  the  inscribed,  escribed, 
and  circumscribed  0. 

65.  How  many  tiles,  6  inches  by  4J  inches,  will  it  take 
to  cover  a  swimming  pool  40  feet  by  27  feet  ? 

66.  Find  the  sides  of  an  isosceles  right  /\  whose  area  is 
98». 


67.  Find  the  area  (in  centares)  and  one  side  of  a  rhom- 
bus, if  the  sum  of  the  diagonals  is  34  feet  and  their  ratio 
is  5  :  12. 


OEOMETMY— NUMERICAL  PROBLEMS.  37 

68.  The  bases  of  a  trapezoid  are  197.3"  and  142.7'°,  and 
its  area  37.57'^ ;  find  its  altitude.     (Log.) 

69.  Find  the  area,  in  square  feet,  of  a  right  /\,  when 
the  sides  are  in  the  ratio  3:4:5,  and  the  altitude  to  the 
hypotenuse  is  1.2^". 

70.  In  the  quadrilateral  A  B  0  D,  A  B  =  10™,  B  C  = 
17^  C  D  =  13°^,  D  A  =  20"°,  and  A  0  =  21'"  ;  find  the 
area  in  hektares,  and  the  perpendiculars  from  B  and  D 
to  AC. 

71.  Find  the  area  of  a  /\  if  the  perimeter  is  82  feet  and 
the  radius  of  the  inscribed  ©1.3  feet. 

72.  Find  the  ratio  of  the  areas  of  two  equilateral  A  if 
the  side  of  one  is  10""  and  the  altitude  of  the  other  is  10'°. 

73.  Find  the  area,  the  altitudes,  and  the  radii  of  the  in- 
scribed, escribed,  and  circumscribed  O  of  the  isosceles  l\ 
whose  leg  is  5  feet  5  inches  and  whose  base  is  10  feet  6 
inches. 

74.  The  bases  of  a  trapezoid  are  13"  and  61"  ;  the  non- 
parallel  sides  are  25"  each  ;  find  the  area  of  the  trapezoid. 

75.  How  many  yards  of  carpet  f  of  a  yard  wide  will  it 
take  to  carpet  a  room  15  feet  by  18  feet  ? 

76.  Find  the  area  of  a  rhombus  whose  perimeter  is  6" 
and  one  of  whose  diagonals  is  1.2". 

77.  The  altitude  of  a  given  A  is  .32^";  find  the  ho- 
mologous altitude,  in  miles,  of  a  similar  l\  49  times  as 
large. 

78.  Find  the  area  of  a  pentagon  whose  perimeter  is 
5.18",  circumscribed  about  a  O  whose  diameter  is  1.1  ". 


38  OEOMETBT— NUMERICAL  PROBLEMS. 

79.  JFind  the  area  in  square  metres  of  a  right  /\  in 
which  a  perpendicular  from  the  vertex  of  the  right  /  to 
the  hypotenuse  divides  the  hypotenuse  into  segments  of 
39H  feet  and  ll^^V  feet.     (Log. ) 

80.  Upon  the  diagonal  of  a  rectangle  6"  by  8"  a  /\  whose 
area  is  three  times  the  area  of  the  rectangle  is  construct- 
ed ;  find  the  altitude  of  the  /\. 

81.  Find  the  side  of  an  equilateral  /\  equivalent  to  the 
sum  of  two  equilateral  A  whose  sides  are  respectively  5"* 
and  12"". 

82.  Find  the  area  of  a  trapezoid  whose  bases  are  26  feet 
and  40  feet,  and  whose  other  sides  are  13  feet  and  15  feet. 

83.  The  three  sides  of  a  A  are  417.31  feet,  589.72  feet, 
and  389.6  feet ;  find  its  area  in  ares.     (Log.) 

84.  Find  the  radii  of  the  inscribed,  escribed,  and  circum- 
scribed ®.     (Log.) 

85.  Find  the  three  altitudes.     (Log.) 

86.  Find  the  median  to  the  longest  side. 

87.  Find  the  bisectors  of  the  three  A.     (Log.) 

88.  The  base  of  a  A  is  25'%  its  altitude  12" ;  find  the 
area  of  the  A  ^^^  ^^  ^J  a  line  parallel  to  the  base  and 
two-thirds  of  the  way  from  the  vertex  to  the  base. 

89.  Two  homologous  sides  of  two  similar  A  are  12  feet 
and  35  feet,  respectively ;  find  the  homologous  side  of  a 
similar  A  equivalent  to  their  sum. 

90.  The  bases  of  a  given  A  and  /  /  are  equal,  and  the 
altitude  of  the  A  is  2™  and  the  altitude  of  the  /  /  5" ; 
find  the  ratio  of  their  areas. 


UNIVERSITY    ! 

OEOMETRT— NUMERICAL  PROBLEMS.  39 

91.  How  many  yards  of  wall  paper  are  required  to  paper 
a  room  25  feet  long,  22  feet  wide,  and  12  feet  high,  allowing 
for  a  chimney  which  projects  into  the  room  1  foot,  one 
door  5  feet  by  7  feet,  another  10  feet  by  10  feet,  a  mantel 
4  feet  by  6  feet,  and  a  window  6  feet  by  11  feet  ? 

92.  The  homologous  altitudes  of  two  similar  A  are  5™ 
and  15"",  respectively ;  what  fraction  of  the  second  is  the 
first? 

93.  Find  the  legs  of  a  right  /\  whose  hypotenuse  is 
25"°"  and  whose  area  is  150"*. 

94.  In  a  /\  whose  base  is  22  feet,  find  the  length  of  the 
line  parallel  to  the  base  and  dividing  the  /\  into  two  equal 
parts.     (Log. ) 

95.  Find  the  area  of  the  /\  whose  sides  are  to  each  other 
as  5  :  12  :  13,  and  whose  altitude  to  the  greater  side  is 
23J  inches. 

96.  The  area  of  the  polygon  Pis  735. 8"^™,  and  of  the 
similar  polygon  Q  is  98.47'*'" ;  find  the  side  of  Q  homolo- 
gous to  a  side  of  P  equal  to  81.41*".     (Log.) 

97.  If  two  sides  of  a  /\  whose  area  is  9  acres  are  165 
rods  and  201  rods,  what  is  the  length  of  the  portions  of 
these  sides  cut  off  by  a  line  parallel  to  the  base  and  cutting 
off  a  /\  of  4  acres  ? 

98.  Find  the  area  of  a  right  /\  whose  hypotenuse  is  70"* 
and  one  of  whose  A  is  60°.     (Log.) 

99.  The  side  of  a  square  is  12" ;  find  the  side  of  a  square 
having  the  ratio  8  to  3  to  this  square. 

100.  In  a  trapezoid  whose  altitude  is  10  feet  and  whose 
bases  are  21  feet  and  29  feet,  what  is  the  length  of  a  line 
parallel  to  the  bases  and  2 J  feet  from  the  smaller  base. 


40  GEOMETRY— NUMERICAL  PROBLEMS, 


BOOK   V. 

Note  I. — The  answers  to  a  large  number  of  the  problems 
of  this  Book  may  be  left  in  an  expressed  form,  if  desired. 
For  example  :  AVhat  is  the  area  of  a  hexagon  inscribed  in  a 

O  whose  radius  is  15  feet  ?    Ans.      ^      v3 

4 

Note  II. — Quite  a  number  of  problems  in  this  Book 
which  seem  difficult,  on  a  mere  reading,  are  rendered  quite 
easy  by  drawing  figures  representing  the  given  conditions 
and  requirements. 

Note  III. — In  many  of  these  problems  it  is  well  to  rep- 
resent the  number  in  terms  of  which  the  answer  is  to  be 
gotten  by  a  letter,  and  then  replace  the  letter  by  its  value 
in  the  final  form  of  the  result,  as  in  finding  the  area,  etc., 
of  circumscribed  and  inscribed  polygons  in  terms  of  the 
radius. 

1.  How  many  degrees  in  each  /  of  a  regular  octagon  ? 
Of  a  regular  dodecagon  ?  Of  a  regular  polygon  of  27 
sides  ? 

2.  How  many  degrees  in  the  /  at  the  centre  of  a  regular 
polygon  of  15  sides  ?     Of  16  sides  ? 

3.  Find  the  side  of  a  square  inscribed  in  a  O  whose  ra- 
dius is  91  feet. 

4.  Find  the  radius  of  a  O  circumscribed  about  a  regular 
hexagon  whose  perimeter  is  5.1™. 

5.  How  many  degrees  in  each  exterior  /  of  a  regular 
polygon  of  18  sides  ?  Of  25  sides  ?  Of  35  sides  ? 

6.  How  many  sides  has  the  regular  polygon  whose  /  at 
the  centre  is  17°  8'  7f' ? 


GEOMETRY— NUMERICAL  PROBLEMS.  41 

7.  How  many  sides  lias  the  regular  polygon  whose  in- 
terior and  exterior  A  are  in  the  ratio  of  18  to  4  ? 

8.  Find  the  side  of  an  equilateral  /\  inscribed  in  a  O 
whose  diameter  is  35.8''™. 

9.  Find  the  perimeter  of  a  regular  decagon  inscribed  in 
a  0  whose  diameter  is  7  feet. 

10.  Find  the  radius  of  the  0  circumscribed  about  a 
regular  hexagon  whose  apothem  is  12-^/3"'";  also  the  area 
of  the  hexagon. 

11.  Find  the  area  of  an  equilateral  /\  inscribed  in  a  O 
whose  radius  is  15  feet.     (Log.) 

12.  Find  the  radius  of  a  0  circumscribed  about  a  square 
whose  area  is  1  square  yard  7  square  feet. 

13.  Find  the  apothem  of  a  regular  hexagon  whose  area 
is  54^3^™. 

14.  Find  the  radius  of  a  O  circumscribed  about  an  equi- 
lateral l\  whose  area  is  27v^3~  square  feet. 

15.  Find  the  area  of  a  regular  hexagon  whose  perimeter 
is  78^"". 

16.  The  apothem  of  an  inscribed  square  is  10  ^/^  ^^et ; 
find  the  area  of  an  equilateral  l\  circumscribed  about  the 
same  0. 

17.  Find  the  area  of  a  regular  polygon  whose  apothem  is 
3.75""",  and  whose  perimeter  is  15"™.  Express  the  result 
in  acres.     (Log.) 

18.  Find  the  side  of  a  regular  decagon  inscribed  in  a  0 
whose  radius  is  35  feet. 


42  GEOMETRY— NUMERICAL  PROBLEMS. 

19.  Find  the  ratio  of  the  areas  of  two  equilateral  A,  one 
inscribed  in,  the  other  circumscribed  about,  a  O  whose  ra- 
dius is  5  inches. 

20.  Find  a  mean  proportional  between  the  areas  of 
problem  19.  Find  the  area  also  of  a  regular  hexagon  in- 
scribed in  the  same  O  (5-inch  radius).  Compare  the  two 
results.     (Log.) 

21.  Find  the  radius  of  a  O  circumscribed  about  a  reg- 
ular hexagon  whose  apothem  is  -— Vs  ^g^^- 


22.  Find  the  area,  in  acres,  of  a  regular  hexagon  cir- 
cumscribed about  a  O  whose  radius  is  7"".     (Log.) 

23.  The  area  of  .an  equilateral  /\  circumscribed  about  a 
given  O  is  87"" ;  find  the  area  of  a  square  inscribed  in  the 
same  O.     (Log.) 

Note. — It  is  customary  to  use  the  value  d\  for  n  in 
problems  involving  English  units,  and  3.1416  where  met- 
ric units  are  employed. 

24.  Find  the  circumference  and  area  of  a  O  whose 
radius  is  11  feet. 

25.  Find  the  diameter  and  area  of  a  O  whose  circum- 
ference is  53f  feet. 

26.  Find  the  circumference  of  a  O  whose  area  is  502,- 
656<=\ 

27.  Two  circumferences  are  in  the  ratio  3  :  5,  and  the 
radius  of  the  larger  is  35°^ ;  what  is  the  radius  of  the 
smaller  ? 

28.  Find  the  radius  of  a  O  equivalent  to  two  ®  whose 
radii  are  respectively  5.6°"*  and  4.2°". 


QEOMETRT— NUMERICAL  PROBLEMS,  43 

29.  What  is  the  length  of  an  arc  of  75°  of  a  O  whose 
radius  is  21  feet  ? 

30.  The  areas  of  two  ©  are  in  the  ratio  of  1 :  5f .  If  the 
radius  of  the  larger  is  4  feet  1  inch,  what  is  the  radius  of 
the  smaller  ? 

31.  Find  the  difference  in  area  between  a  square  and  an 
equilateral  ^  each  inscribed  in  a  O  whose  radius  is  15™. 
(Log-) 

32.  Find  the  area  of  a  segment  of  a  O  of  31-foot  radius 
cut  off  by  the  side  of  a  regular  inscribed  hexagon.    (Log.) 

33.  Find  the  difference  in  length  between  the  circum- 
ference of  a  O  whose  area  is  15836.8056*  and  the  perimeter 
of  the  inscribed  hexagon. 

34.  Find  the  circumference  of  a  O  circumscribed  about 
a  square  field  containing  700  acres.     (Log.) 

35.  Find  the  area  of  a  O  whose  circumference  is 
29.53104°^ 

36.  What  is  the  area  of  a  segment  whose  arc  is  120°,  in 
a  O  whose  radius  is  4. 3"°"  ? 

37.  Find  the  number  of  degrees  in  an  arc  equal  in  length 
to  the  radius  of  its  O. 

38.  What  is  the  ratio  of  the  areas  of  two  O  whose  radii 
are  50  feet  and  65  feet  ? 

39.  Find  the  apothem,  the  side,  and  the  area  of  a  regu- 
lar octagon  inscribed  in  a  O  whose  radius  is  1™.     (Log.) 

40.  How  many  metres  in  the  diameter  of  a  O  whose 
area  is  one  acre  ? 

41.  What  is  the  area  of  a  sector  whose  arc  is  175°  in  a 
O  whose  radius  is  24  feet  ? 


44  GEOMETRY— NUMERICAL  PROBLEMS, 

42.  Find  the  radius  of  a  O  in  which  the  arc  subtended 
by  the  side  of  a  regular  inscribed  dodecagon  is  3.1416'°™. 

43.  How  many  acres  in  a  O,  if  a  quadrant  is  one  mile 
in  length  ? 

44.  AVhat  is  the  ratio  of  the  areas  of  two  ®  whose  cir- 
cumferences are  35™  and  40"°,  respectively  ? 

45.  Find  the  side,  the  apothem,  and  the  area  of  a  regu- 
lar dodecagon  inscribed  in  a  O  whose  diameter  is  3^"". 
(Log.) 

46.  How  far  apart  are  the  circumferences  of  two  con- 
centric O  which  contain  5  acres  and  10  acres,  respective- 
ly?    (Log.) 

47.  Find  the  circumferences  of  the  ®  circumscribed 
about  and  inscribed  in  a  square  whose  side  is  14"".     (Log.) 

48.  Find  the  /  at  the  centre  subtended  by  an  arc  of 
13  inches  in  a  O  whose  radius  is  14^\  inches. 

49.  What  is  the  area  between  three  ®,  each  tangent  to 
the  other  two,  if  each  has  a  radius  of  440  yards  ? 

50.  Find  the  side  of  a  square  equivalent  to  a  O  whose 
radius  is  19  feet. 

51.  Find  the  length  of  a  side  and  the  area  of  a  regular 
octagon  circumscribed  about  a  O  whose  radius  is  a  mile. 
(Log.) 

52.  How  far  apart  are  two  parallel  chords  in  a  O  whose 
radius  is  33  feet,  if  these  chords  are  the  sides  of  regular 
inscribed  polygons,  one  a  hexagon,  the  other  a  dodecagon  ? 
(Log.) 

53.  How  many  rotations  to  the  mile  does  a  wheel  whose 
diameter  is  5  feet  6  inches  make  ? 


OEOMETRT— NUMERICAL  PMOBLEMJS.  45 

54.  Find  the  side  of  a  regular  pentagon  equivalent  to 
the  sum  of  three  regular  pentagons  whose  sides  are  8™, 
9",  and  12'". 

55.  How  much  more  fence  would  it  take  to  enclose  500 
acres  in  the  shape  of  a  square  than  it  would  if  it  were  in 
circular  shape  ? 

56.  Find  the  perimeter  of  a  sector  whose  area  is  77 
square  inches  and  whose  arc  is  45°. 

57.  Find  the  area  of  that  part  of  a  O  whose  radius  is 
7^°"  included  between  two  parallel  chords,  one  of  which  is 
the  side  of  a  regular  inscribed  /\  and  the  other  the  side  of 
an  inscribed  square.     (Log.) 

58.  If  a  bicycle  wheel  makes  680  rotations  to  the  mile, 
what  is  its  diameter  ? 

59.  Find  the  side  and  area  of  a  regular  pentagon  in- 
scribed in  a  O  whose  radius  is  8™. 

60.  Find  the  area  of  a  O  in  which  is  inscribed  a  rectan- 
gle 6  feet  by  8  feet. 

61.  Find  the  area  of  the  regular  hexagon  formed  by 
joining  the  alternate  vertices  of  a  regular  hexagon  whose 
side  is  20  feet. 

62.  Find  the  ratio  of  the  areas  of  the  two  hexagons  in 
problem  61. 

63.  What  is  the  radius  of  a  O  whose  area  is  doubled  by 
increasing  its  radius  7  feet  ? 

64.  Find  the  side  and  the  area  of  a  regular  dodecagon 
circumscribed  about  a  O,  whose  circumference  is  31.416"'°. 
(Log.) 


46  QEOMETRT— NUMERICAL  PROBLEMS. 

65.  Find  the  radius  of  a  O  equivalent  to  three  O, 
whose  diameters  are  54  feet,  56  feet,  and  72  feet. 

66.  What  is  the  difference  in  area  between  an  equilat- 
eral /\  and  a  regular  decagon  each  of  which  has  a  perim- 
eter of  3  miles  ?     (Log.) 

67.  The  area  of  a  segment  cut  off  by  the  side  of  a  regu- 
lar inscribed  hexagon  is  413"* ;  what  is  the  perimeter  of 
this  segment  ?     (Log.) 

68.  Find  the  side  of  a  square  equivalent  to  a  O,  in 
which  a  chord  of  30  feet  has  an  arc  whose  height  is  5  feet. 

69.  Find  the  radius  of  a  O  three  times  as  large  as  a  O 
whose  radius  is  3  feet. 

70.  What  is  the  area  of  a  regular  octagon  whose  perim- 
eter is  28^"'  ?     (Log.) 

71.  Find  the  area  of  the  sector  whose  arc  is  175  feet  in 
a  O  whose  radius  is  133  feet. 

72.  What  must  be  the  width  of  a  walk  which  contains 
1"'  made  around  a  circular  plot  of  ground  containing  5"'  ? 

73.  Find  the  area  of  the  sector  whose  arc  is  the  side  of 
a  regular  inscribed  dodecagon  in  a  O  in  which  a  chord  of 
70  feet  is  12  inches  from  the  centre. 

74.  An  acre  of  ground  lies  between  three  ®,  each  tan- 
gent to  the  other  two  ;  find  the  radius  of  one  of  these  ® . 

75.  Find  the  radius  of  a  O  36  times  as  large  as  a  O 
whose  radius  is  14™. 

76.  If  a  meridian  circle  of  the  earth  is  25,000  miles, 
what  is  the  length  of  the  diameter  in  kilometers  ? 


GEOMETRY— NUMERICAL  PROBLEMS,  47 

77.  If  the  circumference  of  a  Q  is  34.5576^'",  what  is  the 
diameter  of  a  concentric  O  which  divides  it  into  two 
equivalent  parts  ? 

78.  If  the  side  of  a  regular  inscribed  hexagon  cuts  off  a 
segment  whose  area  is  25*,  what  is  the  apothem  of  this 
hexagon  ?     (Log.) 

79.  A  wheel  whose  radius  is  3  feet  6  inches  makes  20 
rotations  per  second  ;  how  many  miles  will  a  point  on  the 
circumference  go  in  a  day  ?     (Log.) 

80.  The  difference  between  the  area  of  a  O  and  its  in- 
scribed square  is  3  acres,  find  the  area  of  the  square  ? 

81.  If  an  8-inch  pipe  will  fill  a  certain  cistern  in  2 
hours  40  minutes,  how  long  will  it  take  a  2-inch  pipe  ? 

82.  Find  the  radius  of  a  O  in  which  an  arc  of  18°  has 
the  same  length  as  an  arc  of  45°  has  in  a  O  whose  radius 
is  56  feet. 

83.  If  the  radius  of  the  earth  is  3,963  miles,  how  many 
metres  is  it  from  the  pole  to  the  equator,  measured  on  a 
meridian  ?     (Log.) 

84.  Upon  each  side  of  a  7-foot  square  as  a  diameter, 
semicircumferences  are  described  within  the  square,  form- 
ing four  leaves,  or  lobes ;  find  the  area  of  one  of  these 
leaves. 

85.  Find  the  number  of  acres  between  two  concentric 
circumferences  which  are  2  miles  and  1  mile  long,  respec- 
tively.    (Log.) 

86.  Find  the  height  of  an  arc  subtended  by  the  side  of 
an  inscribed  dodecagon  in  a  O  whose  area  is  154  square 
feet. 


48  OEOMETBT— NUMERICAL  PROBLEMS, 

87.  Find  the  area  of  a  O  inscribed  in  a  quadrant  of  a 
circle  whose  radius  is  61"°. 

88.  Find  the  area  of  each  part  of  the  quadrant  of  prob- 
lem 87,  outside  the  inscribed  O. 

89.  If  the  circumference  of  a  O,  whose  diameter  is  18™, 
is  divided  into  six  equal  parts,  and  arcs  are  described 
within  the  O,  with  these  points  of  division  as  centres, 
what  is  the  area  of  the  six  leaf -shaped  figures  thus  formed? 

90.  If  a  bridge  in  the  forn^  of  a  circular  arch  18  feet 
high  spans  a  stream  150  feet  wide,  what  is  the  length  of 
the  whole  circumference  of  which  this  arch  is  an  arc  ? 

91.  The  area  inclosed  by  two  tangents  and  two  radii  is 
140»\  If  one  of  the  tangents  =  7"",  find  the  distance 
from  the  centre  to  the  meeting  of  the  tangents  ;  also  the 
area  of  the  O,  in  acres. 

92.  Find  the  sum  of  the  areas  of  the  crescents  formed 
by  describing  semicircumferences  on  the  legs  and  hypot- 
enuse of  a  right  /\  (all  on  one  side),  if  the  legs  are  5  feet 
and  12  feet  respectively.  How  does  this  compare  with  the 
area  of  the  /\  ? 

93.  If  the  sides  of  a  /^  are  40°',  50",  and  60™,  what  is 
the  length  of  the  circumference  of  the  circumscribed  O  ? 

94.  Find  the  sum  of  the  areas  of  two  segments,  cut  off 
by  two  chords,  15  feet  and  20  feet  respectively,  drawn 
from  the  same  point  to  the  extremities  of  the  diameter  of 
their  O. 

95.  If  the  radius  of  the  earth  is  3,963  miles,  how  high 
must  a  light-house  light  be  to  be  seen  30  miles  off  at  sea  ? 


^ 


GEOMETRY— NUMERICAL  PROBLEMS.  49 

96.  The  areas  of  two  concentric  ®  are  to  each  other  as 
5  to  8.  Find  the  radii  of  the  two  ©,  if  the  area  of  that 
part  of  the  ring  which  is  contained  between  two  radii  mak- 
ing the  angle  45°  is  300  square  feet. 

97.  If  two  tangents,  including  an  /  of  60°  and  drawn 
from  the  same  point  without  a  O,  with  two  radii  drawn 
to  their  points  of  contact,  inclose  an  area  of  162^3''", 
find  the  length  of  these  tangents  and  the  area  of  the  sector 
formed  by  these  two  radii  and  their  arc. 

98.  Find  the  area  of  the  segments  of  the  O  in  the  pre- 
ceding problem  made  by  a  chord  perpendicular  to  its 
radius  at  its  middle  point. 

99.  If  a  track,  having  two  parallel  sides  and  two  semi- 
circular ends,  each  equal  to  one  of  the  parallel  sides,  meas- 
ures exactly  a  mile  at  the  curb,  what  distance  does  a  horse 
cover  running  ten  feet  from  the  curb  ?  How  many  acres 
within  the  circuit  he  makes  ? 

100.  Three  ©,  each  tangent  to  the  other  two,  inclose 

with  their  convex  arcs  1"*  of  ground.     How  far  is  it  from 

the  centres  of  these  ®  to  the  middle  point  of  this  piece  of 

ground  ? 

4 


NUMERICAL  PROBLEMS,  EXERCISES,  PROPOSI- 
TIONS,  AND   OTHER  QUESTIONS 

SELECTED  FROM  THE 

ENTRANCE    EXAMINATION    PAPERS    OF    A    NUMBER    OF 
THE  LEADING  COLLEGES  AND   SCIENTIFIC   SCHOOLS. 


1.  From  any  point  in  the  base  of  an  isosceles  triangle 
perpendiculars  are  drawn  to  the  sides ;  prove  their  sum  to 
be  equal  to  the  perpendicular  drawn  from  either  basal  ver- 
tex to  the  opposite  side. — Boston  University. 

2.  The  angle  at  the  vertex  A  of  an  isosceles  triangle 
A  B  C  is  equal  to  twice  the  sum  of  the  equal  angles  B  and 
C.  If  CD  is  drawn  perpendicular  to  BC,  meeting  AB 
produced  at  D,  prove  that  the  triangle  A  C  D  is  equilat- 
eral.—  Wesley  an  University. 

3.  If  from  one  of  the  vertices  (A)  of  a  triangle  (A  B  C) 
a  distance  (AD)  equal  to  the  shorter  one  of  the  two  sides 
(A  B  and  AC)  meeting  in  A  be  cut  off  on  the  longer  one 
(AB),  prove  that  /DCB  =  1  [/AC  B- /AB  C].— C^. 
of  Cal. 

4.  Show  that  the  angle  included  between  the  internal 
bisector  of  one  base  angle  of  a  triangle  and  the  external 
bisector  of  the  other  base  angle  is  equal  to  half  the  verti- 
cal angle  of  the  triangle. — Harvard. 

5.  If  ABC  be  an  equilateral  triangle,  and  if  B  D,  C  D 
bisect  the  angles  B,  C,  the  lines  D  E,  D  F  parallel  to  A  B, 
A  C,  divide  B  C  into  three  equal  parts Cornell. 


QEOMETRT— NUMERICAL  PROBLEMS.  51 

6.  What  is  a  polygon  ?  Prove  that  the  sum  of  the  in- 
terior angles  of  an  n-gon  is  ^  —  2  straight  angles. — Dart- 
mouth. 

7.  A  D  and  B  C  are  the  parallel  sides  of  a  trapezoid 
A  B  C  D,  whose  diagonals  intersect  at  E.  If  F  is  the 
middle  point  of  B  C,  prove  that  E  F  produced  bisects 
AD. — Mass.  Inst.  Tech. 

8.  If  perpendiculars  be  drawn  from  the  angles  at  the 
base  of  an  isosceles  triangle  to  the  opposite  sides,  the  line 
from  the  vertex  to  the  intersection  of  the  perpendiculars 
bisects  the  angle  at  the  vertex  and  the  angle  between  the 
perpendiculars.     Prove Boston  University. 

9.  Prove  that  a  parallelogram  is  formed  by  joining  the 
midpoints  of  the  (adjacent)  sides  of  any  quadrilateral. 
Hint,  draw  the  diagonals  of  the  quadrilateral. — Bowdoiii. 

10.  In  any  triangle  A  B  C,  if  A  D  is  drawn  perpendicu- 
lar to  B  0,  and  A  E  bisecting  the  angle  BAG,  the  angle 
D  A  E  is  equal  to  one-half  the  difference  of  the  angles  B 
and  C. — Cornell. 

11.  Show  that  in  any  right-angled  triangle  the  distance 
from  the  vertex  of  the  right  angle  to  the  middle  point  of 
the  hypotenuse  is  equal  to  one-half  the  hypotenuse. — 
School  of  Mines. 

12.  If  D  is  the  middle  point  of  the  side  B  0  of  the 
triangle  ABC,  and  B  E  and  C  F  are  the  perpendiculars 
from  B  and  0  to  AD,  prove  that  B  E  =  0  F. —  Wesley  an 
University. 

13.  If  in  a  right-angled  triangle  one  of  the  acute  angles 
is  one-third  of  a  right  angle,  the  opposite  side  is  one-half 
the  hypotenuse. —  U.  of  Cal. 

14.  Prove  that  the  diagonals  and  the  line  which  joins 


52  OEOMETBY— NUMERICAL  PROBLEMS. 

the  middle  points  of  the  parallel  sides  of  a  trapezoid  meet 
in  a  point. — Harvard. 

15.  How  many  degrees  in  one  angle  of  an  equiangular 
docedagon  ? — Dartmouth. 

16.  If  the  opposite  sides  of  a  pentagon  be  produced  to 
intersect,  prove  that  the  sum  of  the  angles  at  the  vertices 
of  the  triangles  thus  formed  is  equal  to  two  right  angles. — 
Cornell. 

17.  The  interior  angle  of  a  regular  polygon  exceeds  the 
exterior  angle  by  120°.  How  many  sides  has  the  polygon? 
— Mass.  Inst.  Tech. 

18.  If  one  diagonal  of  a  quadrilateral  bisects  both  angles 
whose  vertices  it  connects,  then  the  two  diagonals  of  the 
quadrilateral  are  mutually  perpendicular.  Prove. — Boston 
University. 

19.  In  a  given  polygon,  the  sum  of  the  interior  angles 
is  equal  to  four  times  the  sum  of  the  exterior.  How  many 
sides  has  the  given  polygon  ? —  Wesley  an  University. 

20.  What  is  the  greatest  number  of  re-entrant  angles  a 
polygon  may  have  compared  to  the  number  of  its  sides  ? 
What  is  the  value  of  the  re-entrant  angles  of  a  pentagon 
in  terms  of  the  interior  angles  not  adjacent  ? — Cornell. 

21.  Show  what  the  sum  of  the  opposite  angles  of  a  quad- 
rilateral inscribed  in  a  circle  is  equal  to.  —  Columbia. 

22.  When  and  why  may  an  arc  be  used  as  the  measure 
of  an  angle  ?  The  vertex  of  an  angle  of  60°  is  outside  a 
circle  and  its  sides  are  secants  ;  what  is  the  relation  be- 
tween the  intercepted  arcs  ? — Dartmouth. 

23.  Show  that  two  angles  at  the  centres  of  unequal 
circles  are  to  each  other  as  their  intercepted  arcs  divided 
by  the  radii. —  U.  of  Cat. 


QEOMETRT— NUMERICAL  PROBLEMS.  63 

24.  Prove  that  in  any  quadrilateral  circumscribed  about 
a  circle  the  sum  of  two  opposite  sides  is  equal  to  the  sum 
of  the  other  two  opposite  sides. — Harvard, 

25.  Construct  a  common  tangent  to  two  circles. — Boston 
University. 

26.  Three  consecutive  sides  of  a  quadrilateral  inscribed 
in  a  circle  subtend  arcs  of  82°,  99°,  and  67°  respectively. 
Find  each  angle  of  the  quadrilateral  in  degrees,  and  the 
angle  between  its  diagonals. —  Yale. 

27.  If  A  C  and  B  C  are  tangents  to  a  circle  whose  centre 
is  0,  from  a  point  C  without  the  circle,  prove  that  the 
centre  of  the  circle  which  passes  through  0,  A,  and  B,  bi- 
sects 0  C. — Mass.  Inst.  Tech. 

28.  Fix  the  position  of  a  given  circle  that  touches  two 
intersecting  lines. —  Vanderhilt  University. 

29.  Through  a  given  point  in  the  circumference  of  a 
circle  chords  are  drawn.  Find  the  locus  of  their  middle 
points. — Cornell. 

30.  Give  contractions  for  the  inscribed,  escribed,  and 
circumscribed  circles  of  any  triangle. — Sheffield  8.  8. 

31.  Construct  a  circle  that  shall  pass  through  two  given 
points  and  shall  cut  from  a  given  circle  an  arc  of  given 
length. —  Vassar. 

32.  Prove  that  the  circumference  of  a  circle  may  be 
passed  through  the  vertices  of  a  quadrilateral  provided  two 
of  its  opposite  angles  are  supplementary. — Boston  Univer- 
sity. 

33.  A  and  B  are  two  fixed  points  on  the  circumference 
of  a  circle,  and  P  Q  is  any  diameter.  What  is  the  locus 
of  the  intersection  of  P  A  and  Q  B  ? — Harvard. 

34.  The  length  of  the  straight  line  joining  the  middle 


54  GEOMETRY— NUMERICAL  PROBLEMS. 

points  of  the  non-parallel  sides  of  a  circumscribed  trapezoid 
is  equal  to  one-fourth  the  perimeter  of  the  trapezoid. — 
Mass.  Inst.  Tech. 

35.  The  points  of  tangency  of  a  quadrilateral,  circum- 
scribed about  a  circle,  divide  the  circumference  into  arcs, 
which  are  to  each  other  as  4,  6,  10,  and  16.  Find  the  an- 
gles of  the  quadrilateral. — Harvard. 

36.  Given  three  indefinite  straight  lines  in  the  same 
plane,  no  two  of  which  are  parallel,  show  that  four  circles 
can  be  described  to  touch  the  three  lines. 

If  two  of  the  three  lines  are  parallel,  show  that  the  four 
circles  reduce  to  two. — Cornell. 

37.  From  a  fixed  point  0  of  a  given  circumference  are 
drawn  two  chords,  OP,  0  Q,  so  as  to  make  equal  angles 
with  a  fixed  chord,  0  R,  between  them.  Prove  that  P  Q 
will  have  the  same  direction  whatever  the  magnitude  of  the 
angles. — Harvard. 

38.  Draw  a  straight  line  tangent  to  a  given  circle  and 
parallel  to  a  given  straight  line. —  Yale. 

39.  Given  two  parallel  lines  and  a  secant  line,  also  two 
circles  each  tangent  to  both  parallels  and  to  the  secant ; 
prove  that  the  distance  between  the  centres  equals  the 
segment  of  the  secant  line  intercepted  between  the  two 
parallels. — Boston  University. 

40.  The  vertices  of  a  quadrilateral  inscribed  in  a  circle 
divide  the  circumference  into  arcs  which  are  to  each  other 
as  1,  2,  3,  and  4.  Find  the  angles  between  the  opposite 
sides  of  the  quadrilateral. — Harvard. 

41.  Show  how  to  construct  an  isosceles  triangle  with  a 
given  base  and  a  given  vertical  angle. — School  of  Mines. 

42.  Two  circumferences  intersect  at  A  and  B.  Through 
B  any  secant  is  drawn  so  as  to  cut  the  circumferences  in  0 


GEOMETRY— NUMERICAL  PROBLEMS.  55 

and  D  respectively.  Show  that  the  angle  0  A  D  is  the  same 
for  all  secants  drawn  through  B.  What  value  has  this  an- 
gle when  the  circumferences  intersect  each  other  orthogo- 
nally ? — Harvard. 

43.  The  perimeter  of  the  circumscribed  equilateral  tri- 
angle is  double  that  of  the  similar  inscribed  triangle. — 
Sheffield  S.  S. 

44.  The  radius  of  a  circle  is  13  inches.  Through  a  point 
6  inches  from  the  centre  a  chord  is  drawn.  What  is  the 
product  of  the  two  segments  of  the  chord  ?  What  is  the 
length  of  the  shortest  chord  that  can  be  drawn  through  that 
point  ? — Wesley  an  University. 

45.  A  B  is  the  hypotenuse  of  a  right  triangle  ABC.  If 
perpendiculars  be  drawn  to  A  B  at  A  and  B,  meeting  A  C 
produced  at  D,  and  B  0  produced  at  E,  prove  the  triangles 
A  C  E  and  B  0  D  similar.— FaZe. 

46.  Prove  that  the  diagonal  of  a  square  is  incommensur- 
able with  its  side.  When  are  two  quantities  said  to  be  in- 
commensurable ? — Boiudoin. 

47.  A  B  C  D  is  an  inscribed  quadrilateral.  The  sides  A  B 
and  D  C  are  produced  to  meet  at  E.  Prove  triangles  ACE 
and  BDE  similar. — Mass.  Inst.  Tech. 

48.  A  chord  18  inches  long  is  bisected  by  another  chord 
22  inches  long.  Find  the  segments  of  the  latter. — N.  J. 
State  College. 

49.  In  any  given  triangle,  if  from  two  of  the  vertices 
perpendiculars  be  drawn  to  the  opposite  sides,  the  triangle 
cut  off  by  the  line  joining  the  feet  of  the  perpendiculars  is 
similar  to  the  given  triangle. —  U.  of  Cat. 

50.  The  diagonals  of  a  certain  trapezoid,  which  are  8  and 
12  feet  long  respectively,  divide  each  other  into  segments 


66  OEOMETRY— NUMERICAL  PROBLEMS. 

which  in  the  case  of  the  shorter  diagonal  are  3  feet  and  5 
feet  long.  What  are  the  segments  of  the  other  diagonal  ? — 
Harvard. 

51.  The  sides  of  a  triangle  are  5,  6,  and  8.  Find  the  seg- 
ments of  the  last  side  made  by  a  perpendicular  from  the 
opposite  angle. — Rutgers  8.  8. 

52.  In  a  plane  triangle  what  is  the  square  on  the. side 
opposite  to  the  obtuse  angle  equal  to  ?  Demonstrate. — 
8chool  of  Mines. 

53.  The  sides  of  a  triangle  are  9,  8,  13.  Is  the  greatest 
angle  acute,  obtuse,  or  right  ? —  Vassar. 

54.  Given  A  B  =  xy,  write  five  resulting  proportions. 
Need  not  prove. — Boston  University. 

55.  The  radii  of  two  circles  are  8  inches  and  3  inches, 
and  the  distance  between  their  centres  is  15  inches.  Find 
the  length  of  their  common  tangents Wesleyan  Univer- 


56.  The  bases  of  two  similar  triangles  are  respec- 
tively 12.34  and  18.14  metres.  The  altitude  of  the  first  is 
6.12  metres  ;  find  the  altitude  of  the  second.  (Use  loga- 
rithms.)— Yale. 

57.  If  A  B  and  0  D  are  equal  chords  of  a  circle  and  in- 
tersect at  E,  prove  that  A  E  =  E  D  and  B  E  =  E  Q.—Mass. 
Inst.  Tech. 

58.  One  segment  of  a  chord  drawn  through  a  point  7 
units  from  the  centre  of  a  circle  is  4  units.  If  the  diame- 
ter of  the  circle  is  15  units,  what  is  the  other  segment  ?— 
Brown. 

59.  Two  parallel  chords  of  a  circle  are  d  and  h  in  length, 
and  their  distance  apart  is/;  what  is  the  radius  ? — Van- 
derhilt  University. 


GEOMETRY— NUMERICAL  PROBLEMS.  57 

60.  In  a  certain  circle  a  chord  is  10  inches  long,  while 
another  chord  twice  as  far  from  the  centre  as  the  first  is  5 
inches  long  ;  find  the  radius  of  the  circle  and  the  distances 
of  the  chords  from  the  centre. — Harvard. 

61.  When  is  a  line  said  to  be  divided  harmonically  9 
From  the  point  P  without  a  circle  a  secant  through  the 
centre  is  drawn  cutting  the  circle  in  A  and  B.  Tangents 
are  drawn  from  P  and  the  points  of  contact  connected  by 
a  line  cutting  A  B  in  Q.  Show  that  P  and  Q  divide  A  B 
harmonically. — Sheffield  8.  8. 

62.  Two  sides  of  a  triangle  are  17  and  10  ;  the  perpen- 
dicular from  their  intersection  to  the  third  side  is  8  ;  what 
is  the  length  of  the  third  side  ? — Mass.  Inst.  Tech. 

63.  Prove  that  the  sum  of  the  squares  of  the  sides  of  a 
parallelogram  is  equal  to  the  sum  of  the  squares  of  its  diag- 
onals.— School  of  Mines. 

64.  In  a  triangle  whose  sides  are  48,  36,  and  50,  where 
do  the  bisectors  of  the  angles  intersect  the  sides  ?  What 
are  the  lengths  of  the  bisectors  ? — Rutgers  8.  8. 

65.  The  distance  from  the  centre  of  a  circle  to  a  chord 
10  inches  long  is  12  inches.  Find  the  distance  from  the 
centre  to  a  chord  24  inches  long. —  Wesleyan  University. 

66.  The  diameter  of  a  circle  is  20  inches,  the  least  dis- 
tance from  a  certain  point  upon  the  circumference  to  a 
diameter  is  8  inches  ;  find  the  distances  from  this  point  to 
the  ends  of  the  above  diameter. — Boston  University. 

67.  Let  A  B  C  be  a  right  triangle.  The  two  sides  about 
the  right  angle  C  are  respectively  455  and  1,092  feet.  The 
hypotenuse  A  B  is  divided  into  two  segments  A  E  and  B  E 
by  the  perpendicular  upon  it  from  0.  Compute  the  lengths 
of  A  E,  B  E,  and  C  l^.—Yale. 


58  GEOMETRY— NUMERICAL  PROBLEMS. 

68.  0  is  any  point  on  the  straight  portion,  A  B,  of  the 
boundary  of  a  semicircle.  C  T>,  drawn  at  right  angles  to 
A  B,  meets  the  circumference  at  D.  D  0  is  drawn  to  the 
centre,  0,  of  the  circle,  and  the  perpendicular  dropped 
from  C  upon  0  D  meets  0  D  at  E.  Show  that  D  0  is  a 
mean  proportional  to  A  0  and  D  E. — Harvard. 

69.  The  length  of  one  side  of  a  right  triangle  is  12,  and 
the  length  of  the  perpendicular  from  its  extremity  to  the 
hypotenuse  is  4y\.  Find  the  lengths  of  hypotenuse  and 
other  side. — Mass.  Inst.  Tech. 

70.  The  three  sides  of  a  triangle  are  6,  8,  10  units  long  ; 
compute  the  lengths  of  the  three  medial  lines. — Cornell. 

71.  The  area  of  a  rectangle  is  64,  the  difference  of  two 
adjacent  sides  is  12  ;  construct  the  rectangle. — Bowdoin. 

72.  Prove  that  if  any  point  on  one  of  the  diagonals  of  a 
parallelogram  be  joined  to  the  vertices,  of  the  triangles 
thus  formed,  those  having  the  same  base  are  equivalent. — 
U.  of  Gal. 

73.  In  a  triangle  ABO,  let  0  be  the  point  in  which  the 
medians  (lines  drawn  from  the  vertices  to  the  middle  points 
of  the  opposite  sides)  intersect.  Prove  that  the  triangles 
OAB,  OAO,  OBC  are  equivalent. — Amherst. 

74.  If  two  equivalent  triangles  have  a  common  base,  and 
lie  on  opposite  sides  of  it,  the  base,  or  the  base  produced, 
will  bisect  the  line  joining  the  vertices. — Dartmouth. 

75.  If  the  perimeter  of  a  rectangle  is  72  feet,  and  the 
length  is  equal  to  twice  the  width,  find  the  area. — Johns 
Hopkins  University. 

76.  The  area  of  a  certain  isosceles  triangle  is  50  square 
feet,  and  each  of  its  equal  sides  is  10  feet  long ;  find  the 
angles  of  the  triangle. — Cornell, 


GEOMETRY— NUMERICAL  PROBLEMS,  59 

77.  Two  mutually  equiangular  triangles  are  similar. 
The  base  of  a  triangle  is  32  feet,  its  altitude  20  feet.  What 
is  the  area  of  the  triangle  cut  off  by  drawing  a  line  parallel 
to  the  base  and  at  a  distance  of  15  feet  from  the  base  ? — 
Wesleyan  University. 

78.  The  perimeter  of  a  trapezoid  is  56  inches.  If  each 
of  the  non-parallel  sides  is  13  inches  long,  and  the  area  is  180 
square  inches,  what  are  the  respective  lengths  of  the  parallel 
sides? — Mass.  Inst.  Tech. 

79.  The  area  of  a  certain  polygon  is  5  square  feet.  Find 
the  area  of  a  similar  polygon  whose  perimeter  is  in  the 
ratio  of  M  to  N  to  that  of  the  given  polygon. — Sheffield 

S.  S. 

80.  A  vertex  of  a  parallelogram  and  the  middle  points 
of  the  two  sides  adjacent  to  it  form  the  vertices  of  a  trian- 
gle whose  area  is  equal  to  one-eighth  the  area  of  the  paral- 
lelogram.— Boston  University. 

81.  (a.)  If  two  triangles  are  on  equal  bases  and  between 
the  same  parallels,  a  line  parallel  to  their  bases  cuts  off 
equal  areas. 

{jb.)  Lines  joining  the  non-adjacent  extremities  of 

two  parallel  chords  are  equal, 
(c.)  State  and  prove   the  converse  of  the  preceding 

proposition. —  Yale. 

2      X 

82.  Given -  =  -•   Construct  x. — Cornell. 

X      6 

83.  Find  the  area  of  a  triangle  in  terms  of  its  sides. — 
Vanderhilt  University. 

84.  Prove  that,  if  in  the  triangle  ABC  the  line  drawn 
from  the  vertex  C  to  the  middle  point  of  the  opposite  side 
is  equal  to  half  the  latter,  the  area  of  the  triangle  is  nu- 


60  OEOMETRY— NUMERICAL  PROBLEMS. 

merically  equal  to  half  the  product  of  A  C  by  B  C. — Har- 
vard. 

85.  Given  three  rectangles,  find  a  square  whose  area  is 
equal  to  the  sum  of  the  areas  of  the  larger  two  minus  the 
area  of  the  smallest  one. —  U.  of  Cal. 

86.  Prove  that  the  square  described  upon  the  altitude  of 
an  equilateral  triangle  has  an  area  three  times  as  great  as 
that  of  a  square  described  upon  half  of  one  side  of  the  tri- 
angle.—  Cornell. 

87.  A  D  and  B  C  are  the  parallel  sides  of  the  trapezoid 
A  B  C  D,  whose  diagonals  intersect  at  0.     Prove 

area  A 0 D  :  area  B  0  C  =  aO^  '-  00^- 

— Mass.  Inst.   Tech. 

88.  Construct  a  square  whose  area  is  3  times  that  of  a 
given  square. — Sheffield  8.  8. 

89.  Draw  a  hexagon  having  one  re-entrant  angle,  and 
construct  a  triangle  equivalent  to  this  polygon. — Cornell. 

90.  The  parallel  sides  of  a  trapezoid  are  12  and  18,  the 
non-parallel  sides  are  each  5  ;  find  its  area  and  the  altitude 
of  the  triangle  formed  by  producing  the  non-parallel  sid.es 
until  they  meet. — Dartmouth.  ' 

91.  Through  a  point  in  one  side  of  a  triangle  draw  a 
line  parallel  to  the  base  which  shall  bisect  the  area  of  the 
triangle. — Cornell. 

92.  The  area  of  a  polygon  is  160  square  feet,  one  side 
is  6  feet  long  ;  find  the  homologous  side  of  a  similar  poly- 
gon whose  area  is  800  square  feet. — Boston  U^iiversit^^ 

93.  The  base  of  a  triangle  is  16  feet,  and  the  two  other 
sides  are  respectively  12  and  10  feet.  Find  the  altitude  of 
the  triangle,  and  also  the  area. — Yale. 


GEOMETRT—NUMERIGAL  PROBLEMS.  61 

94.  In  a  certain  triangle  ABC,  AO'-BO^  =  iAB^; 
show  that  a  perpendicular  dropped  from  0  upon  A  B  will 
divide  the  latter  into  segments  which  are  to  each  other  as 
3  to  1. —  Harvard. 

95.  Construct  a  parallelogram  equivalent  to  a  given  tri- 
angle and  having  one  of  the  diagonals  equal  to  a  given 
line. —  U.  of  Cal. 

96.  Construct  a  polygon  similar  to  a  given  polygon  and 
having  two  and  a  half  times  its  area. — Cornell. 

97.  How  many  degrees  in  each  angle  of  a  regular  deca- 
gon ? — Yale. 

98.  If  the  diagonals  A  C  and  B  Gr  of  the  regular  octa- 
gon ABCDEFGH  intersect  at  0,  how  many  degrees 
are  there  in  the  angle  AOB  ? — Mass.  hist.  Tech. 

99.  Show  that  the  sum  of  the  alternate  angles  of  an  in- 
scribed hexagon  (not  necessarily  regular)  is  equal  to  four 
right  angles. — School  of  Mines. 

100.  An  equilateral  triangle  is  inscribed  in  a  circle. 
Find  its  side,  apothem,  and  area  in  terms  of  the  radius  R. 
— Dartmouth. 

101.  Find  the  ratio  of  the  area  of  a  regular  hexagon  in- 
scribed in  a  circle  to  that  of  a  regular  hexagon  circum- 
scribed about  the  same  circle. — Johns  Hopkins  University. 

102.  What  regular  polygon  has  each  angle  equal  to  five 
thirds  of  a  right  angle  ? —  U.  of  Cal. 

103.  A  certain  equilateral  triangle  has  sides  8  V  3  inches 
long  ;  what  is  the  radius  of  the  circumference  circumscribed 
about  this  triangle  ? — Harvard. 

104.  Compute  the  area  of  a  regular  hexagon  whose  side 
is  5  feet.  Construct  a  triangle  of  equivalent  area. — Sheffield 
S.  S. 


62  QEOMETRT— NUMERICAL  PROBLEMS. 

105.  The  area  of  the  regular  inscribed  hexagon  of  a  cir- 
cle is  three-fourths  of  that  of  the  regular  circumscribed 
hexagon . — Cor7iell. 

106.  Find  the  number  of  degrees  in  an  angle  of  a  regu- 
lar pentagon  and  give  proof  of  the  process. — Boiodoin. 

107.  If  the  interior  angles  of  any  quadrilateral  be  bisected 
and  each  bisector  produced  to  meet  two  others,  the  quadri- 
lateral formed  may  be  inscribed  in  a  circle.  Prove. — Bos- 
ton Univei'sity. 

108.  The  diagonals  of  a  regular  pentagon  divide  each 
other  in  mean  and  extreme  ratio. —  U.  of  Cal. 

109.  Show  that  an  equiangular  polygon  inscribed  in  a  cir- 
cle is  regular  if  the  number  of  its  sides  is  odd. — Cornell. 

110.  The  radius  of  a  certain  circle  is  9  inches  ;  find  the 
area  of  that  one  of  all  the  regular  polygons  inscribed  in  it 
which  has  the  shortest  perimeter.  How  long  a  perimeter 
can  a  regular  polygon  inscribed  in  this  circle  have  ? — Har- 
vard. 

111.  A  regular  hexagon,  ABCDEF,  is  inscribed  in  a 
circle  whose  radius  is  2  ;  find  the  length  of  the  diagonal 
AC. — Mass.  Inst.  Tech. 

112.  To  compute  the  area  of  a  circle  whose  radius  is 
unity. — Dartmouth. 

113.  Find  the  area  of  a  circle  inscribed  in  a  square  con- 
taining 400  square  feet. — N.  J.  State  College. 

114.  Find  the  side  of  a  square  equivalent  to  a  circle 
whose  radius  is  56  feet.     (Use  logarithms.) — Yale. 

115.  The  area  of  a  certain  regular  hexagon  is  294  V  3 
square  inches  ;  find  the  area  and  the  circumference  of  the 
circumscribed  circle. — Harvard. 


GEOMETRY— NUMERICAL  PROBLEMS.  63 

116.  The  circumference  of  a  circle  is  78.54  inches  ;  find 
(1)  its  diameter,  and  (2)  its  area. — Rutgers  S.  S. 

117.  If  the  areas  of  two  regular  pentagons  be  as  16  to  25, 
and  the  perimeter  of  the  first  pentagon  be  50  inches,  what 
is  the  perimeter  of  the  second  ? — Cornell. 

118.  If  the  radius  of  a  circle  is  5,  find  the  area  of  the 
sector  whose  central  angle  is  50°. —  Wesleyan  University. 

119.  The  angle  of  a  sector  is  30°  ;  the  radius  is  12.  Find 
the  area  of  the  sector. — Amherst. 

120.  Prove  that  the  area  of  the  regular  inscribed  dode- 
cagon is  equal  to  three  times  the  square  of  the  radius. — 
U.  of  Cat. 

121.  If  the  diameter  of  a  circle  is  3  inches,  what  is  the 
length  of  an  arc  of  80°  ? — Mass.  Inst.  Tech. 

122.  In  a  circle  whose  radius  is  8,  what  is  the  length  of 
the  arc  of  a  sector  of  45**  ?  What  is  the  area  of  this  sec- 
tor ? — Rutgers  S.  S. 

123.  If  the  radius  of  a  circle  is  5  inches,  compute  its  cir- 
cumference and  its  area  ;  also  the  perimeter,  the  area,  and 
the  apothem  of  an  inscribed  square. —  Yale. 

124.  The  perimeter  of  a  regular  hexagon  is  480  feet,  and 
that  of  a  regular  octagon  is  the  same.  Which  is  the  greater 
in  area,  and  by  how  much  ? — Cornell. 

125.  The  area  of  a  certain  circle  is  154  square  inches  ; 
what  angle  at  the  centre  is  subtended  by  an  arc  of  the  cir- 
cumference 5  J  inches  long  ? — Harvard. 

126.  Find  the  length  of  the  arc  of  75°  in  the  circle 
whose  radius  is  5  feet. — iV.  /.  State  College. 

127.  A  M  and  B  N  are  perpendiculars  from  points  A  and 
B  to  the  line  M  N.     Find  a  point  P  on  the  line  M  N  such 


64  GEOMETRY— NUMERICAL  PROBLEMS. 

that  the  sum  of  the  distances  A  P,  B  P,  is  the  least  pos- 
sible.—  Wellesley. 

128.  Two  circles  are  tangent  internally,  the  ratio  of 
their  radii  being  2  : 3.  Compare  their  areas,  and  also  the 
area  left  in  the  larger  circle  with  each. — Sheffield  8.  8. 

129.  A  kite-shaped  racing-track  is  formed  by  a  circular 
arc  and  two  tangents  at  its  extremities.  The  tangents 
meet  at  an  angle  of  60°.  The  riders  are  to  go  round  the 
track,  one  on  a  line  close  to  the  inner  edge,  the  other  on  a 
line  everywhere  5^  ft.  outside  the  first  line.  Show  that 
the  second  rider  is  handicapped  by  about  22  feet. — Har- 
vard. 

130.  The  diameters  of  two  water-pipes  are  6  and  8  inches 
respectively.  What  is  the  diameter  of  a  pipe  having  a 
capacity  equal  to  their  sum  ? — Rutgers  8.  8. 

131.  (a.)  There  are  two  gardens:  one  is  a  square  and 
the  other  a  circle  ;  and  they  each  contain  a  hectare.  How 
much  farther  is  it  around  one  than  the  other  ? 

(b.)  If  the  area  of  each  is  2  hectares,  what  will  be  the 
difference  of  their  perimeters  ? —  Yale. 

132.  Inscribe  a  square  in  a  scalene  triangle. — Cornell. 

133.  A  horse  is  tethered  to  a  hook  on  the  inner  side  of  a 
fence  which  bounds  a  circular  grass-plot.  His  tether  is  so 
long  that  he  can  just  reach  the  centre  of  the  plot.  The 
area  of  so  much  of  the  plot  as  he  can  graze  over  is  -*/ 
(4  TT  —  3  a/3)  sq.  rd.  ;  find  the  length  of  the  tether  and 
the  circumference  of  the  plot. — Harvard. 

134.  If  the  apothem  of  a  regular  hexagon  is -2,  find  the 
area  of  its  circumscribed  circle. —  Wesley  an  University. 

135.  Of  all  polygons  formed  of  given  sides  the  maxi- 
mum may  be  inscribed  in  a  circle. — Sheffield  8.  8. 


OEOMETRT— NUMERICAL  PROBLEMS.  65 

136.  If  the  radius  of  a  circle  is  6,  what  is  the  area  of  a 
segment  whose  arc  is  60°  ?  (Take  n  =  3.1416.) — Mass. 
Inst.  Tech. 

137.  A  stone  bridge  20  ft.  wide  has  a  circular  arch  of 
140  ft.  span  at  the  water  level.  The  crown  of  the  arch  is 
140  {1  —  i  a/3)  ft.  above  the  surface  of  the  water.  How 
many  square  feet  of  surface  must  be  gone  over  in  cleaning 
so  much  of  the  under  side  of  the  arch  as  is  above  water  ? — 
Harvard. 

138.  Of  all  isoperimetric  figures  the  circle  has  the 
greatest  area. — Corfiell. 

139.  Compute  by  logarithms  the  value  of 


s/ 


(2.3456)3  X  (.301456)^ 

(4.02356)^  —Yale. 


SELECTED  EXAMINATION  PAPERS  IN  PLANE 
GEOMETRY  SET  FOR  ADMISSION  TO  A  NUMBER 
OF  THE  LEADING  COLLEGES  AND  SCIENTIFIC 
SCHOOLS  IN  THE    UNITED   STATES. 

Harvard,  June,  1892. 

[In  solving  probleniB  use  for  n  the  approximate  value  Sf .] 

1.  Prove  that  if  two  sides  of  a  triangle  are  unequal,  the 
angle  opposite  the  greater  side  is  greater  than  the  angle  op- 
posite the  less  side. 

In  a  certain  right  triangle  one  of  the  legs  is  half  as  long  as 
the  hypotenuse  ;  what  are  the  angles  of  the  triangle  ? 

2.  Show  how  to  find  on  a  given  indefinitely  extended  straight 
line  in  a  plane,  a  point  O  which  shall  be  equidistant  from  two 
given  points  A,  B  in  the  plane.  If  A  and  B  lie  on  a  straight 
line  which  cuts  the  given  line  at  an  angle  of  45°  at  a  point 
7  inches  distant  from  A  and  17  inches  from  B,  show  that  O  A 
will  be  13  inches. 

3.  Prove  that  an  angle  formed  by  a  tangent  and  a  chotd 
drawn  through  its  point  of  contact  is  the  supplement  of  any 
angle  inscribed  in  the  segment  cut  off  by  the  chord.  What  is 
the  locus  of  the  centre  of  a  circumference  of  given  radius 
which  cuts  at  right  angles  a  given  circumference  ? 

4.  Show  that  the  areas  of  similar  triangles  are  to  each  other 
as  the  squares  of  the  homologous  sides. 

5.  Prove  that  the  square  described  upon  the  altitude  of  an 
equilateral  triangle  has  an  area  three  times  as  great  as  that  of 
a  square  described  upon  half  of  one  side  of  the  triangle. 

6.  Find  the  area  included  between  a  circumference  of 
radius  7  and  the  square  inscribed  within  it. 


GEOMETRY— NUMERICAL  PROBLEMS.  67 


Harvard,  June,  1893. 

[In  solving  problems  use  for  v  the  approximate  value  3f .] 

1.  Prove  that  two  oblique  lines  drawn  from  a  given  point 
to  a  given  line  are  equal  if  they  meet  the  latter  at  equal  dis- 
tances from  the  foot  of  the  perpendicular  dropped  from  the 
point  upon  it. 

How  many  lines  can  be  drawn  through  a  given  point  in  a 
plane  so  as  to  form  in  each  case  an  isosceles  triangle  with  two 
given  lines  in  the  plane  ? 

3.  Prove  that  in  the  same  circle,  or  in  equal  circles,  equal 
chords  are  equally  distant  from  the  centre,  and  that  of  two 
unequal  chords  the  less  is  at  the  greater  distance  from  the 
centre. 

Two  chords  of  a  certain  circle  bisect  each  other.  One  of  them 
is  10  inches  long ;  how  far  is  it  from  the  centre  of  the  circle  ? 

A  variable  chord  passes,  when  produced,  through  a  fixed 
point  without  a  given  circle.  What  is  the  locus  of  the  mid- 
dle point  of  the  chord  ? 

3.  A  common  tangent  of  two  circumferences  which  touch 
each  other  externally  at  A,  touches  the  two  circumferences  at 
B  and  C  respectively  ;  show  that  B  A  is  perpendicular  to  A  C. 

4.  Assuming  that  the  areas  of  two  triangles  which  have  an 
angle  of  the  one  equal  to  an  angle  of  the  other  are  to  each 
other  as  the  products  of  the  sides  including  the  equal  angles, 
prove  that  the  bisector  of  an  angle  of  a  triangle  divides  the 
opposite  side  into  parts  which  are  proportional  to  the  sides 
adjacent  to  them. 

5.  Prove  that  the  circumferences  of  two  circles  have  the 
same  ratio  as  their  radii. 

6.  A  quarter-mile  running  track  consists  of  two  parallel 
straight  portions  joined  together  at  the  ends  by  semicircum- 
ferences.  The  extreme  length  of  the  plot  enclosed  by  the 
track  is  180  yards.  Find  the  cost  of  sodding  this  plot  at  a 
quarter  of  a  dollar  per  square  yard. 


68  OEOMETRT— NUMERICAL  PROBLEMS. 


Harvard,  June,  1894. 

[In  solving  problems  use  for  n  the  approximate  value  3f .] 

1.  Prove  that  any  quadrilateral  the  opposite  sides  of  which 
are  equal,  is  a  parallelogram. 

A  certain  parallelogram  inscribed  in  a  circumference  has 
two  sides  20  feet  in  length  and  two  sides  15  feet  in  length ; 
what  are  the  lengths  of  the  diagonals  ? 

2.  Prove  that  if  one  acute  angle  of  a  triangle  is  double  an- 
other, the  triangle  can  be  divided  into  two  isosceles  triangles 
by  a  straight  line  drawn  through  the  vertex  of  the  third 
angle. 

Upon  a  given  base  is  constructed  a  triangle  one  of  the  base 
angles  of  which  is  double  the  other.  The  bisector  of  the 
larger  base  angle  meets  the  opposite  side  at  the  point  P.  Find 
the  locus  of  P. 

3.  Show  how  to  find  a  mean  proportional  between  two  given 
straight  lines,  but  do  not  prove  that  your  construction  is  cor- 
rect. 

Prove  that  if  from  a  point,  O,  in  the  base,  B  C,  of  a  triangle, 
ABC,  straight  lines  be  drawn  parallel  to  the  sides,  A B, 
A  C,  respectively,  so  as  to  meet  A  C  in  M  and  A  B  in  N,  the 
area  of  the  triangle  A  M  N  is  a  mean  proportional  between  the 
areas  of  the  triangles  B  N  O  and  C  M  O. 

4.  Assuming  that  the  areas  of  two  parallelograms  which 
have  an  angle  and  a  side  common  and  two  other  sides  unequal, 
but  commensurable,  are  to  each  other  as  the  unequal  sides, 
prove  that  the  same  proportion  holds  good  when  these  sides 
have  no  common  measure. 

5.  Every  cross-section  of  the  train-house  of  a  railway  station 
has  the  form  of  a  pointed  arch  made  of  two  circular  arcs  the 
centres  of  which  are  on  the  ground.  The  radius  of  each  arc  is 
equal  to  the  width  of  the  building  (210  feet)  ;  find  the  dis- 
tance across  the  building  measured  over  the  roof,  and  show 
that  the  area  of  the  cross-section  is  3, 675  (4  n-  —  3  4/3  )  square 
feet. 


GEOMETRY— NUMERICAL  PROBLEMS.  69 


Harvard,  June,  1895. 

One  question  may  he  omitted. 
[In  solving  problems  use  for  n  the  approximate  value  3f  ] 

1.  Prove  that  if  two  straight  lines  are  so  cut  by  a  third  that 
corresponding  alternate-interior  angles  are  equal,  the  two 
lines  are  parallel  to  each  other. 

2.  Prove  that  an  angle  formed  by  two  chords  intersecting 
within  a  circumference  is  measured  by  one-half  the  sum  of 
the  arcs  intercepted  between  its  sides  and  between  the  sides  of 
its  vertical  angle. 

Two  chords  which  intersect  within  a  certain  circumference 
divide  the  latter  into  parts  the  lengths  of  which,  taken  in 
order,  are  as  1,  1,  2,  and  5  ;  what  angles  do  the  chords  make 
with  each  other  ? 

3.  Through  the  point  of  contact  of  two  circles  which  touch 
each  other  externally,  any  straight  line  is  drawn  terminated 
by  the  circumferences  ;  show  that  the  tangents  at  its  extrem- 
ities are  parallel  to  each  other. 

What  is  the  locus  of  the  point  of  contact  of  tangents 
drawn  from  a  fixed  point  to  the  different  members  of  a  system 
of  concentric  circumferences  ? 

4.  Prove  that,  if  from  a  point  without  a  circle  a  secant  and 
a  tangent  be  drawn,  the  tangent  is  a  mean  proportional  be- 
tween the  whole  secant  and  the  part  without  the  circle. 

Show  (without  proving  that  your  construction  is  correct) 
how  you  would  draw  a  tangent  to  a  circumference  from  a 
point  without  it. 

5.  Prove  that  the  area  of  any  regular  polygon  of  an  even 
number  of  sides  (2  n)  inscribed  in  a  circle  is  a  mean  propor- 
tional between  the  areas  of  the  inscribed  and  the  circum- 
scribed polygons  of  half  the  number  of  sides.  If  n  be  indef- 
initely increased  what  limit  or  limits  do  these  three  areas  ap- 
proach ? 


70  GEOMETRT—NUMERIGAL  PROBLEMS. 


6.  The  perimeter  of  a  certain  church 
window  is  made  up  of  three  equal  semi- 
circumferences,  the  centres  of  which  form 
the  vertices  of  an  equilateral  triangle 
which  has  sides  3i  feet  long.  Find  the 
area  of  the  window  and  the  length  of  its 
perimeter. 


Harvard,  June,  1896. 

One  question  may  he  omitted. 
[In  solving  problems  use  for  n-  the  approximate  value  3^.] 

1.  Prove  that  if  two  oblique  lines  drawn  from  a  point  to  a 
straight  line  meet  this  line  at  unequal  distances  from  the  foot 
of  the  perpendicular  dropped  upon  it  from  the  given  point, 
the  more  remote  is  the  longer. 

2.  Prove  that  the  distances  of  the  point  of  intersection  of 
any  two  tangents  to  a  circle  from  their  points  of  contact  are 
equal. 

A  straight  line  drawn  through  the  centre  of  a  certain  circle 
and  through  an  external  point,  P,  cuts  the  circumference  at 
points  distant  8  and  18  inches  respectively  from  P.  What  is 
the  length  of  a  tangent  drawn  from  P  to  the  circumference  ?. 

3.  Given  an  arc  of  a  circle,  the  chord  subtended  by  the  arc 
and  the  tangent  to  the  arc  at  one  extremity,  show  that  the 
perpendiculars  dropped  from  the  middle  point  of  the  arc  on 
the  tangent  and  chord,  respectively,  are  equal. 

One  extremity  of  the  base  of  a  triangle  is  given  and  the 
centre  of  the  circumscribed  circle.  What  is  the  locus  of  the 
middle  point  of  the  base  ? 

4.  Prove  that  in  any  triangle  the  square  of  the  side  opposite 
an  acute  angle  is  equal  to  the  sum  of  the  squares  of  the  other 
two  sides  diminished  by  twice  the  product  of  one  of  those 
sides  and  the  projection  of  the  other  upon  that  side. 


GEOMETRY— NUMERICAL  PROBLEMS.  71 

Show  very  briefly  how  to  construct  a  triangle  liaving  given 
the  base,  the  projections  of  the  other  sides  on  the  base,  and 
the  projection  of  the  base  on  one  of  these  sides. 

5.  Show  that  the  areas  of  similar  triangles  are  to  one  an- 
other as  the  areas  of  their  inscribed  circles. 

The  area  of  a  certain  triangle  the  altitude  of  which  is  >/  2, 
is  bisected  by  a  line  drawn  parallel  to  the  base.  What  is  the 
distance  of  this  line  from  the  vertex  ? 

6.  Two  flower-beds  have  equal  perimeters.  One  of  the  beds 
is  circular  and  the  other  has  the  form  of  a  regular  hexagon. 
The  circular  bed  is  closely  surrounded  by  a  walk  7  feet 
wide  bounded  by  a  circumference  concentric  with  the  bed. 
The  area  of  the  walk  is  to  that  of  the  bed  as  7  to  9.  Find 
the  diameter  of  the  circular  bed  and  the  area  of  the  hexagonal 
bed. 


Yale,  June,  1892. 

TIME  ALLOWED,    ONE  HOUR. 

1.  Construct  accurately,  by  ruler  and  compass,  a  parallelo- 
gram A  B  C  D  having  the  angle  A  45°,  the  side  A  B  6  units  in 
length,  and  the  altitude  3  of  the  same  units. 

Calculate  the  length  of  A  C. 

2.  (a)  State  the  converse  of  the  following  proposition  : 

//  a  triangle  is  isosceles  and  if  a  straight  line  is  drawn 
through  the  vertex  parallel  to  the  base,  it  bisects  an  exterior 
angle  of  the  triangle. 

(6)  Prove  the  converse  as  you  have  stated  it. 

Make  the  demonstration  as  full  and  clear  as  possible. 

3.  Prove  two  of  the  following  propositions  :  The  work  may 
be  limited  to  drawing  a  figure  and  giving  a  synopsis  of  the 
demonstration. 


72  QEOMETBY— NUMERICAL  PROBLEMS. 

(a)  If  the  area  of  a  regular  polygon  is  equal  to  the  product  of 
the  perimeter  by  one-half  the  apothegm,  it  follows  that  the  area 
of  a  oirale  =  n  R^. 

(&)  If  two  lines  are  drawn  through  the  same  point  across  a 
circle,  the  products  of  the  two  distances  on  each  line  from  this 
point  to  the  circumference  are  equal  to  each  other. 

(c)  If  the  radius  of  a  circle  he  dimded  in  extreme  and  mean 
ratio,  the  greater  segment  is  equal  to  one  side  of  a  regular  in- 
scribed decagon. 


Yale,  June,  1893. 

1.  Prove  that  if  the  diagonals  of  a  quadrilateral  bisect  each 
other  the  figure  is  a  parallelogram. 

3.  Prove  that  in  any  right-angled  triangle  the  square  on  the 
side  opposite  to  the  right  angle  is  equal  to  the  sum  of  the 
squares  on  the  other  two  sides.  • 

A  purely  geometrical  proof  is  preferred. 

State  fully  each  principle  employed  in  the  proof. 

3.  Given  a  straight  line  AB,  of  indefinite  length,  and  a 
point  C  without  it.  Find  a  point  in  A  B  equally  distant  from 
A  and  C. 

Make  the  necessary  construction  accurately  with  ruler  amd 
compass. 

In  what  case  is  the  solution  impossible  ? 

4.  Given  an  angle  C  O  D  at  the 
centre  of  a  circle  and  the  line  C  A 
meeting  D  O  produced  in  A  so 
that  A  B  is  equal  to  the  radius  of 
the  circle.  Prove  that  the  angle 
A  is  equal  to  one-third  of  the 
angle  COD. 


OEOMETBY— NUMERICAL  PROBLEMS,  73 

Yale,  June,  1894. 
GEOMETRY  (A). 

TIME  ONE  HOUR. 

1.  If  the  diagonals  of  a  quadrilateral  bisect  each  other,  the 
figure  is  a  parallelogram. 

2.  To  draw  a  tangent  to  a  given  circle,  so  that  it  shall  be 
parallel  to  a  given  straight  line. 

3.  If  A  B  is  a  chord  of  a  circle,  and  C  E  is  any  chord  drawn 
through  the  middle  point  C  of  the  arc  A  B  cutting  the  chord 
A  B  at  D,  prove  that  the  chord  A  C  is  a  mean  proportional 
between  C  D  and  C  E. 

4.  The  areas  of  two  similar  triangles  are  to  each  other  as 
the  squares  of  any  two  homologous  sides. 

5.  The  area  of  a  circle  is  equal  to  one-half  the  product  of  its 
circumference  and  radius. 


Yale,  June,  1894. 
GEOMETRY  (B). 

TIME  FORTY-FIVE  MIISTJTES. 

1.  What  is  the  number  of  degrees  in  each  angle  of  a  regu- 
lar decagon  ? 

2.  Find  the  area  in  square  feet  of  an  equilateral  triangle 
whose  side  is  3  metres. 

3.  A  B  C  is  a  right  triangle.  The  sides  A  C  and  B  C  about 
the  right  angle  C  are  respectively  50  and  120  feet.  Divide  the 
triangle  into  two  parts  equal  in  area  by  a  line  D  F  parallel  to 
B  C.  Compute  the  length  of  the  three  sides  of  the  triangle 
ADF. 


74  QEOMETRT—NUMERIGAL  PROBLEMS, 

4.  The  area  of  a  circle  is  a  hectare.     What  is  its  diameter  ? 

5.  Calculate  in  metres  the  length  of  a  degree  on  the  circum- 
ference of  the  earth,  assuming  the  section  of  the  earth  to  be  a 
circle  whose  radius  is  3, 963  miles.  [Those  taking  the  prelimi- 
nary examinations  must  use  logarithms.  ] 

[For  preliminary  candidates  only.] 

6.  Find  the  value  of  the  following  expression  by  logarithms : 


^- 


(.06342)^  X  187.32 
.34216  X  6.0372 


Yale,  June,  1895. 
GEOMETRY   (A). 

TIME  AliLOWED,    SIXTY  MIN^UTES. 

1.  (a)  Define  the  terms  "  locus  "  and  *'  limit  of  a  variable  " 
and  give  an  example  of  each. 

(&)  Prove  that  two  triangles  are  similar  if  their  homologous 
sides  are  proportional. 

(c)  Through  a  given  point  A  within  a  circle  draw  two  equal 
chords. 

[Both  the  construction  (with  ruler  and  compass),  and  also 
the  proof,  are  required.  ] 

Prove  that  if  each  of 
two  angles  of  a  quadri- 
lateral is  a  right  angle, 
the  bisectors  of  the  other 
angles  are  either  perpen- 
dicular, or  parallel,  to 
each  other. 

(6)  Prove  that  if  the  radius  of  a  circle  is  divided  in  extreme 
and  mean  ratio,  the  greater  part  is  equal  to  the  side  of  a  regu- 
lar inscribed  decagon. 

[The  construction  is  not  required.} 


QEOMETBT— NUMERICAL  PROBLEMS.  75 

Yale,  June,  1896. 
GEOMETRY   (B). 

TIME  ALLOWED,    FORTY-FIVE  MINUTES. 

One  question  may  he  omitted.     Logarithmic  tables  should  he 
u^sed  in  calculating  the  answers  of  two  questions. 

1.  The  base  of  a  triangle  is  14  inches  and  its  altitude  is  7 
inches.  Find  the  area  of  the  trapezoid  cut  off  by  a  line  6 
inches  from  the  vertex. 

Express  the  result  in  square  metres. 

2.  Find  the  number  of  feet  in  an  arc  of  40°  12'  if  the  radius 
of  the  circle  is  0.7539  metres. 

3.  The  length  of  a  chord  is  10  feet,  and  its  greatest  dis- 
tance from  the  subtending  arc  is  3  feet  7i  inches.  Find  the 
radius  of  the  circle. 

4.  Find  the  area,  and  also  the  weight  in  grams,  of  the 
largest  square  that  can  be  cut  from  a  circular  sheet  of  tin  16 
inches  in  diameter  and  weighing  8.2  ounces  per  square  foot. 


Yale,  June,  1896. 
GEOMETRY  (A). 

TIME,    ONE  HOUR. 


1.  The  sum  of  the  three  angles  of  a  triangle  is  equal  to  two 
right  angles. 

2.  Construct  a  circle  having  its  centre  in  a  given  line  and 
passing  through  two  given  points. 

3.  The  bisector  of  the  angle  of  a  triangle  divides  the  op- 
posite side  into  segments  which  are  proportional  to  the  two 
other  sides. 


76  GEOMETRY— NUMERICAL  PROBLEMS. 

4.  If  two  angles  of  a  quadrilateral  are  bisected  by  one  of  its 
diagonals,  the  quadrilateral  is  divided  into  two  equal  tri- 
angles and  the  two  diagonals  of  the  quadrilateral  are  per- 
pendicular to  each  other. 

5.  The  circumferences  of  two  circles  are  to  each  other  as 
their  radii.     (Use  the  method  of  limits.) 


Yale,  June,  1896. 
GEOMETRY  (B). 

TIME  ALLOWED,    FORTY-FIVE   MINUTES. 

1.  A  tree  casts  a  shadow  90  feet  long,  when  a  vertical  rod  6 
feet  high  casts  a  shadow  4  feet  long.     How  high  is  the  tree  ? 

2.  The  distance  from  the  centre  of  a  circle  to  a  chord  10 
inches  long  is  12  inches.  Find  the  distance  from  the  centre 
to  a  chord  24  inches  long. 

3.  The  diameter  of  a  circular  grass  plot  is  28  feet.  Find 
the  diameter  of  a  grass  plot  just  twice  as  large.  (Use  loga- 
rithms. ) 

4.  Find  the  area  of  a  triangle  whose  sides  are  a  =  12.342 
metres  h  =  31.456  metres  o  =  24.756  metres,  using  the  formula 

^      ^  a  +  b  +  c  ' 

Area  =  ^  s{s-a)  («-&)  >^^  where  s=  — ^ (Use  lo- 
garithms.) 

Princeton,  June,  1894. 

What  text-book  have  you  read  ? 

1.  Prove  that  the  sum  of  the  three  angles  of  a  triangle  is 
equal  to  two  right  angles.  Define  triangle,  right  angle,  right 
triangle,  scalene  triangle. 

2.  Prove  that  the  opposite  sides  and  angles  of  a  parallelo- 
gram are  equal.     Define  a  parallelogram,  a  rectangle. 


OEOMETRT-—NUMERIGAL  PROBLEMS.  77 

3.  Prove  that  an  angle  inscribed  in  a  circle  is  measured  by 
one-half  of  the  arc  intercepted  by  its  sides. 

Consider  all  cases. 

4.  Show  how  to  construct  a  triangle,  having  given  two 
sides  and  the  angle  opposite  one  of  them. 

Is  the  construction  always  possible?  If  not,  state  when 
and  why  it  fails. 

5.  Prove  that  if  any  chord  is  drawn  through  a  fixed  point 
within  a  circle,  the  product  of  its  segments  is  constant  in 
whatever  direction  the  chord  is  drawn. 

6.  Prove  the  ratio  between  the  areas  of  two  triangles  which 
have  an  angle  of  the  one  equal  to  an  angle  of  the  other. 

Define  area. 

7.  Define  a  regular  polygon  and  prove  that  two  regular 
polygons  of  the  same  number  of  sides  are  similar. 

Define  similar  figures. 


Princeton,  June,  1895. 

What  text-book  have  you  read  ? 

1.  Prove  that  every  point  in  a  perpendicular  erected  at  the 
middle  of  a  given  straight  line  is  equidistant  from  the  extrem- 
ities of  the  line,  and  every  point  not  in  the  perpendicular  is 
unequally  distant  from  the  extremities  of  the  line. 

2.  Prove  that  the  sum  of  the  interior  angles  of  a  polygon  is 
equal  to  two  right  angles  taken  as  many  times  less  two  as  the 
figure  has  sides. 

Define  a  polygon,  also  a  right  angle. 

3.  Prove  that  the  tangents  to  a  circle  drawn  from  an  ex- 
terior point  are  equal,  and  make  equal  angles  with  the  secant 
drawn  from  this  point  through  the  centre  ;  also  that  either 
tangent  is  a  mean  proportional  between  the  secant  and  its 
external  segment. 

Define  circle,  tangent,  secant,  chord,  mean  proportional. 


V 


NIVERSITY 

or 
r-. 


78  QEOMETRT—NUMERIOAL  PROBLEMS. 

4.  Show  how  to  circumscribe  a  circle  about  a  given  triangle, 
giving  reasons  for  the  process. 

5.  Prove  what  the  area  of  a  triangle  is  equal  to  ;  also  the 
area  of  a  trapezoid. 

Define  triangle,  trapezoid,  area. 

6.  Prove  that  the  area  of  a  circle  is  equal  to  one-half  the 
product  of  the  circumference  by  the  radius. 

Express  the  area  of  a  circle  in  terms  of  tt. 
Define  n  and  give  its  numerical  value. 


Princeton,  June,  1896. 

State  what  text-book  you  have  read  and  how  much  of  it. 

1.  Prove  that  the  sum  of  the  three  angles  of  a  triangle  is 
equal  to  two  right  angles ;  and  that  the  sum  of  all  the  in- 
terior angles  of  a  polygon  of  n  sides  is  equal  to  {n — 2)  times  two 
right  angles. 

2.  Show  that  the  portions  of  any  straight  line  intercepted 
between  the  circumferences  of  two  concentric  circles  are  equal. 

3.  Define  similar  polygons  and  show  that  two  triangles 
whose  sides  are  respectively  parallel  or  perpendicular  are  simi- 
lar polygons  according  to  the  definition. 

4.  Prove  that,  if  from  a  point  without  a  circle  a  secant 
and  a  tangent  are  drawn,  the  tangent  is  a  mean  proportional 
between  the  whole  secant  and  its  external  segment. 

5.  Prove  what  the  area  of  a  triangle  is  equal  to  ; — also  of  a 
trapezoid  ; — also  of  a  regular  polygon.  Define  each  of  the  fig- 
ures named. 

6.  Explain  how  to  construct  a  triangle  equivalent  to  a  given 
polygon. 

7.  Prove  that  of  all  isoperimetric  polygons  of  the  same 
number  of  sides,  the  maximum  is  equilateral. 


OEOMETRY—NUMERIOAL  PROBLEMS.  79 

Princeton,  September,  1896. 
State  what  text-book  you  have  read  and  how  much  of  it. 

1.  Name  and  define  six  quadrilateral  figures. 

Prove  that  in  a  parallelogram  the  opposite  sides  are  equal, 
and  the  diagonals  bisect  each  other. 

2.  Define  and  show  how  to  construct  the  inscribed  circle  and 
the  three  escribed  circles  of  a  given  triangle. 

3.  Prove  that,  if  the  base  of  a  triangle  is  divided,  either  in- 
ternally or  externally,  into  segments  proportional  to  the  other 
two  sides,  the  line  joining  the  point  of  section  and  the  oppo- 
site vertex  of  the  triangle  is  the  bisector  of  the  angle  (either 
internal  or  external)  at  that  vertex. 

4.  Prove  what  the  area  of  a  parallelogram  is  equal  to,  and 
show  how  to  construct  a  square  equivalent  to  a  given  paral- 
lelogram. 

5.  Prove  that  if  a  circle  is  divided  into  any  number  of  equal 
parts,  the  chords  joining  the  successive  points  of  division  form 
a  regular  inscribed  polygon,  and  the  tangents  drawn  at  the 
points  of  division  form  a  regular  circumscribed  polygon. 

6.  Prove  that  the  maximum  of  all  isoperimetric  polygons 
of  the  same  number  of  sides  is  a  regular  polygon. 


Columbia,  June,  1896. 

TIME  ALLOWED,    TWO  AND   ONE-HALF  HOURS. 

Omit  one  question  from  each  of  the  groups,  A,  B,  C. 
State  what  text-book  you  have  used  in  preparation. 

A. 

1.  Prove  that,  in  a  circle,  a  diameter  is  greater  than  any 
other  chord. 

2.  Prove  that,  in  any  triangle,  a  line  drawn  parallel  to  the 
base  divides  the  other  sides  proportionally. 


80  GEOMETRY— NUMERICAL  PROBLEMS. 

3.  Prove  that  an  angle  formed  by  a  tangent  and  a  chord  of 
a  circle  meeting  at  the  point  of  contact  of  the  tangent,  is 
measured  by  one-half  of  the  included  arc. 

B. 

4.  Prove  that  if  four  quantities  are  in  proportion,  they  are 
in  proportion  by  composition  and  by  division. 

5.  Show  how  to  construct  a  triangle  equal  to  a  given  pen- 
tagon. 

6.  Show  how  to  inscribe  a  regular  decagon  in  a  circle. 


C. 

7.  Let  A,  B,  C,  D  be  four  points  lying  in  the  order  named 
upon  a  certain  circumference.  The  arcs  A  B,  B  C,  and  C  D, 
are  of  76°,  53°,  and  118°  respectively.  Find  the  angle  between 
the  chords  A  C  and  B  D,  and  also  the  angle  between  A  B  and 
C  D,  produced. 

8.  Prove  that  the  difference  of  the  diagonals  of  any  quadri- 
lateral is  less  than  the  sum  of  either  pair  of  opposite  sides. 

9.  Find  a  point  in  the  base  of  a  triangle  such  that  lines 
drawn  from  it  parallel  to  the  other  side  of  the  triangle  shall 
be  equal  to  each  other. 


School  of  Mines,  June,  1896. 

TIME  ALLOWED,    TWO  AND  ONE-HALF  HOURS. 

1.  Prove  that  if  a  straight  line,  E  F,  has  two  of  its  points, 
E  and  F,  each  equally  distant  from  two  points,  A  and  B,  it  is 
perpendicular  to  the  line  A  B  at  its  middle  point. 

2.  In  equal  circles  incommensurable  angles  at  the  centre  are 
proportional  to  their  intercepted  arcs  :  demonstrate. 

3.  In  the   parallelogram   A  B  C  D   straight  lines  join  the 


GEOMETRY— NUMERICAL  PROBLEMS,  81 

middle  point  E  of  side  B  C  with  the  vertex  A,  and  the  middle 
point  F  of  side  A  D  with  the  vertex  C.  Show  that  A  E  and  F  C 
are  parallel  and  that  the  diagonal  B  D  is  trisected . 

4.  Show  that  the  areas  of  similar  triangles  are  to  each  other 
as  the  squares  of  their  homologous  sides. 

5.  How  do  you  divide  a  line  in  extreme  and  mean  ratio  ? 

6.  What  are  the  immediate  propositions  which  lead  up  to 
the  determination  of  the  area  of  the  circle  of  radius  unity,  and 
how  is  this  area  determined?  No  demonstrations  are  re- 
quired. 


University  of  Pennsylvania,  June,  1893. 

TWO  HOURS. 

1.  If  two  straight  lines  intersect  each  other,  the  opposite 
(or  vertical)  angles  are  equal. 

The  straight  lines  which  bisect  a  pair  of  adjacent  angles 
formed  by  two  intersecting  straight  lines  are  perpendicular 
to  each  other. 

2.  If  each  side  of  a  polygon  is  extended,  the  sum  of  the  ex- 
terior angles  is  four  right  angles. 

3.  In  the  same  circle,  or  in  equal  circles,  equal  chords  are 
equally  distant  from  the  centre,  and  of  two  unequal  chords, 
the  less  is  at  the  greater  distance  from  the  centre. 

The  least  chord  that  can  be  drawn  in  a  circle  through  a 
given  point  is  the  chord  perpendicular  to  the  diameter  through 
the  point. 

4.  Two  triangles  are  similar  when  they  are  mutually  equi- 
angular, 

5.  Show  how  to  find  a  mean  proportional  between  two 
given  lines. 

6 


82  GEOMETRY— NUMERICAL  PROBLEMS. 

6.  The  square  described  upon  the  hypotenuse  of  a  right- 
angled  triangle  is  equivalent  to  the  sum  of  the  squares  de- 
scribed upon  the  other  two  sides.  {Give  the  pure  geometric 
proof.) 

7.  In  a  triangle  any  two  sides  are  reciprocally  proportional 
to  the  perpendiculars  let  fall  upon  them  from  the  opposite 
vertices. 

8.  The  area  of  the  regular  inscribed  triangle  is  half  the 
area  of  the  regular  inscribed  hexagon. 


University  of  Pennsylvania,  June,  1895.    v 

TIME  :  ONE  HOUR  AND  A  HALF. 

Give  all  the  work. 

1.  The  interior  and  exterior  bisectors  of  any  angle  of  a  tri- 
angle divide  the  opposite  side  into  segments  which  are  pro- 
portional to  the  adjacent  sides. 

2.  If  two  of  the  medial  lines  of  a  triangle  are  equal,  the  tri- 
angle is  an  isosceles. 

3.  The  area  of  a  rhombus  is  240  and  its  side  is  17,  find  its 
diagonals. 

4.  Construct  a  square  whose  area  shall  be  five  times  the  area 
of  a  given  square. 

5.  The  parallelogram  formed  by  lines  joining  the  middle 
points  of  the  adjacent  sides  of  a  quadrilateral  is  equivalent  to 
one-half  the  quadrilateral. 

6.  If  the  interior  bisector  of  the  angle  C  and  the  exterior  bi- 
sector of  the  angle  B  of  a  triangle  ABC  meet  at  D,  prove 
that  angle  B  D  C  =  i  A. 

7.  In  any  triangle  the  product  of  two  sides  is  equal  to  the 
diameter  of  the  circumscribed  circle  multiplied  by  the  per- 
pendicular to  the  third  side  from  its  opposite  vertex. 


GEOMETRY— NUMERICAL  PROBLEMS.  83 

8.  Define  tt.     Give  a  method  for  computing  an  approximate 
value  of  IT. 

9.  If  the  radius  of  a  circle  is  r,  what  is  the  side  of  the  in- 
scribed decagon  ? 


TTniversity  of  Pennsylvania,  September,  1895. 

TIME  :    ONE   HOUR  AND  A  HALF. 

Give  all  the  work. 

1.  The  lines  joining  the  middle  points  of  the  adjacent  sides 
of  any  quadrilateral  form  a  parallelogram  whose  perimeter  is 
equal  to  the  sum  of  the  diagonals  of  the  quadrilateral. 

2.  Prove  that  the  bisectors  of  the  angles  of  a  rectangle  form 
a  square. 

3.  The  three  medial  lines  of  a  triangle  intersect  in  one  point 
which  divides  each  medial  line  in  the  ratio  1  :  3. 

4.  If  from  a  point  a  tangent  and  a  secant  to  a  circle  are 
drawn,  the  tangent  is  a  mean  proportional  between  the  whole 
secant  and  its  external  segment. 

5.  Similar  triangles  are  to  each  other  as  the  squares  of  two 
homologous  sides. 

6.  Divide  a  given  straight  line  in  extreme  and  mean  ratio. 

7.  Construct  a  triangle  which  shall  be  similar  to,  and  three 
times  as  large  as,  a  given  triangle. 

8.  From  a  given  point  without  a  circle  draw  a  secant  whose 
external  and  internal  segments  shall  be  equal. 

9.  If  the  radius  of  a  circle  is  2,  what  is  the  area  of  a  sector 
whose  central  angle  is  152°  ? 


84  OEOMETRY— NUMERICAL  PROBLEMS. 

University  of  Pennsylvania,  June,  1896. 

TIME  :    TWO   HOURS. 

1.  Define  :  Altitude  of  a  triangle,  medial  line,  regular  poly- 
gon, inscribed  angle,  segment  and  sector  of  a  circle. 

2.  If  two  parallels  are  cut  by  a  straight  line,  the  alternate 
exterior  angles  are  equal. 

3.  Either  side  of  a  triangle  is  greater  than  the  difference  of 
the  other  two. 

4.  The  sum  of  the  angles  of  any  polygon  is  equal  to  twice  as 
many  right  angles  as  the  polygon  has  sides,  less  four  right 
angles. 

5.  The  areas  of  similar  triangles  are  to  each  other  as  the 
squares  of  their  homologous  sides. 

6.  The  lines  joining  the  middle  points  of  the  sides  of  any 
quadrilateral  is  a  parallelogram. 

7.  Construct  a  square  equivalent  to  a  given  triangle. 

8.  The  line  joining  the  middle  points  of  the  two  non-paral- 
lel sides  of  a  trapezoid  is  12^  inches,  the  distance  between  the 
parallel  sides  is  8|  inches,  what  is  the  side  of  a  regular  hexa- 
gon equivalent  to  the  trapezoid  ? 

9.  Define  n.     Outline  a  method  for  computing  it. 


University  of  Pennsylvania,  September,  1896. 

TIME  :  TWO  HOURS. 

1.  Define  :  An  angle  (right,  acute,  and  obtuse),  tangent  to  a 
circle,  regular  polygon,  mention  all  different  kinds  of  paral- 
lelograms. 

2.  If  two  straight  lines  are  cut  by  a  third,  making  the  al- 
ternate-interior angles  equal,  the  two  sides  are  parallel. 


GEOMETRY— NUMEBIGAL  PROBLEMS.  86 

3.  In  any  triangle  the  greater  angle  lies  opposite  the  greater 
side. 

4.  What  is  each  angle  in  a  regular  pentagon,  regular  hexa- 
gon, regular  dodecagon  ? 

5.  If  in  a  right  triangle  a  perpendicular  be  drawn  from  the 
vertex  of  the  right  angle  to  the  hypotenuse,  the  perpendicular 
is  a  mean  proportional  between  the  segments  of  the  hypote- 
nuse. 

6.  The  lines  joining  the  middle  points  of  the  sides  of  a  rhom- 
bus form  a  rectangle. 

7.  Construct  a  square  equivalent  to  a  given  pentagon. 

8.  The  base  of  a  triangle  is  7.345  inches  and  the  altitude 
4.756  inches,  what  is  the  side  of  a  regular  triangle  which  has 
the  same  area  as  the  given  triangle  ? 

9.  Find  the  area  of  a  regular  hexagon  inscribed  in  a  circle 
whose  radius  is  11.529  inches. 


Cornell,  1894. 


1.  If  two  triangles  have  two  sides  of  the  one  equal,  respec- 
tively, to  two  sides  of  the  other,  but  the  included  angle  of 
the  first  greater  than  the  included  angle  of  the  second,  then 
the  third  side  of  the  first  is  greater  than  the  third  side  of  the 
second.     Prove  this  ;  and  state  the  converse. 

2.  Prove  that  lines  drawn  through  the  vertices  of  a  triangle 
to  the  middle  points  of  the  opposite  sides  meet  in  a  point. 

How  do  the  areas  of  the  three  triangles  formed  by  joining 
this  point  to  the  vertices  of  the  original  triangle  compare  ? 
Why? 

3.  If  equilateral  triangles  be  constructed  upon  each  side  of 
any  given  triangle,  prove  that  the  lines  drawn  from  their 
outer  vertices  to  the  opposite  vertices  of  the  given  triangle 
are  equal. 


86  GEOMETRY— NUMERICAL  PROBLEMS. 

4.  From  any  point  P,  outside  of  a  circle  whose  centre  is  at 
O,  two  tangents  are  drawn  touching  the  circle  at  A  and  B  ; 
at  Q,  a  variable  point  in  the  smaller  arc  AB,  a  tangent  is 
drawn  cutting  the  other  two  tangents  in  H  and  K.  Prove 
that  the  perimeter  of  the  triangle  P  H  K  is  constant,  and  also 
that  the  angle  H  O  K  is  constant.  Compare  this  angle  with 
the  angle  P. 

5.  If  similar  parallelograms  be  described  upon  the  three 
sides  of  a  right  triangle  as  homologous  sides,  prove  that  the 
parallelogram  described  upon  the  hypotenuse  is  equivalent  to 
the  sum  of  those  described  upon  the  other  two  sides. 

6.  Prove  that  the  sum  of  the  perpendiculars  drawn  to  the 
sides  of  a  regular  polygon  from  any  point  P  within  the  figure, 
is  equal  to  the  apothem  of  the  polygon  multiplied  by  the 
number  of  its  sides. 

State  this  proposition,  so  modified,  that  the  point  P  may 
be  without  the  polygon. 

7.  Of  all  isoperimetric  triangles  having  the  same  base,  that 
which  is  isosceles  has  the  maximum  area. 


Cornell,  1895. 

One  question  may  he  omitted. 

1.  The  sum  of  the  lines  which  join  a  point  within  a  triangle 
to  the  three  vertices  is  less  than  the  perimeter,  but  greater 
than  half  the  perimeter. 

2.  Two  triangles  are  equal  if  the  three  sides  of  one  are  equal 
respectively  to  the  three  sides  of  the  other. 

3.  Construct  through  a  point,  P,  exterior  to  a  circle,  a  secant 
P A B  so  that  AB^"  =  P A   x  P B. 

4.  The  radius  of  a  circle  is  6  inches  ;  through  a  point  10 
inches  from  the  centre  tangents  are  drawn.     Find  the  lengths 


GEOMETRY— JSrUMERIGAL  PROBLEMS.  87 

of  the  tangents,  also  of  the  chord  joining  the  points  of  con- 
tact. 

5.  Construct  a  polygon  similar  to  two  given  similar  poly- 
gons, and  equivalent  to  their  sum. 

6.  The  bisector  of  an  angle  of  a  triangle  divides  the  opposite 
side  into  segments  proportional  to  the  other  two  sides. 

7.  The  perimeter  of  an  inscribed  equilateral  triangle  is  equal 
to  half  the  perimeter  of  the  circumscribed  equilateral  triangle. 

8.  If  one  of  the  acute  angles  of  a  right  triangle  is  double  the 
other,  the  hypotenuse  is  double  the  shorter  side. 


Johns  Hopkins  University,  October,  1896. 

1.  Prove  that  the  bisectors  of  the  two  pairs  of  vertical  an- 
gles formed  by  two  intersecting  lines  are  perpendicular  to  each 
other. 

2.  Show  that  through  three  points  not  lying  in  the  same 
straight  line  one  circle,  and  only  one,  can  be  made  to  pass. 

3.  The  bases  of  a  trapezoid  are  16  feet  and  10  feet  respec- 
tively ;  each  leg  is  5  feet.  Find  the  area  of  the  trapezoid. 
Also  find  the  area  of  a  similar  trapezoid,  if  each  of  its  legs  is 
3  feet. 

4.  Define  regular  polygon.  Prove  that  every  equiangular 
polygon  circumscribed  about  a  circle  is  a  regular  polygon. 

5.  Prove  that  the  opposite  angles  of  a  quadrilateral  in- 
scribed in  a  circle  are  supplements  of  each  other. 

6.  Construct  a  square,  having  given  its  diagonal. 

7.  Prove  that  the  area  of  a  triangle  is  equal  to  half  the 
product  of  its  perimeter  by  the  radius  of  the  inscribed  circle. 

8.  What  is  the  area  of  the  ring  between  two  concentric 
circumferences  whose  lengths  are  10  feet  and  20  feet  respec- 
tively ? 


88  aEOMETRT— NUMERICAL  PROBLEMS, 


Sheffield  Scientific  School,  June,  1892. 

[Note.— state  at  the  head  of  your  paper  what  text-book  you  have  studied  on  the  sub- 
ject and  to  what  extent.] 

1.  Prove  the  two  propositions  relating  to  the  sum  of  the  in- 
terior angles  of  a  convex  polygon,  and  the  sum  of  the  exterior 
angles  formed  by  producing  each  side  in  one  direction. 

2.  In  a  circle  the  greater  chord  subtends  the  greater  arc, 
and  conversely. 

3.  When  is  a  line  said  to  be  divided  harmonically  f  From 
the  point  P  without  a  circle  a  secant  through  the  centre  is 
drawn  cutting  the  circle  in  A  and  B.  Tangents  are  drawn 
from  P  and  the  points  of  contact  connected  by  a  line  cutting 
A  B  in  Q.     Show  that  P  and  Q  divide  A  B  harmonically. 

4.  Derive  an  expression  for  the  area  of  a  regular  polygon. 

5.  When  two  sides  of  a  triangle  are  given  at  what  angle  must 
fchey  intersect  if  the  area  shall  be  maximum  ?  Prove  your 
answer. 


Sheffield  Scientific  School,  June,  1896. 

[NoTB.— state  at  the  head  of  your  paper  what  text-book  you  have  studied  on  the  sub- 
ject and  to  what  extent.] 

1.  Two  angles  whose  sides  are  parallel  each  to  each  are  either 
equal  or  supplementary.  When  will  they  be  equal,  and  when 
supplementary  ? 

3.  An  angle  formed  by  two  chords  intersecting  within  the 
circumference  of  a  circle  is  measured  by  one-half  the  sum  of 
the  intercepted  arcs. 

3.  A  triangle  having  a  base  of  8  inches  is  cut  by  a  line  par- 
allel to  the  base  and  6  inches  from  it.     If  the  base  of  the 


GEOMETRY— NUMERICAL  PROBLEMS.  89 

smaller  triangle  thus  formed  is  5  inches,  find  the  area  of  the 
larger  triangle. 

4.  Construct  a  parallelogram  equivalent  to  a  given  square, 
having  given  the  sum  of  its  base  and  altitude.     Give  proof. 

5.  What  are  regular  polygons?  A  circle  may  be  circum- 
scribed about,  and  a  circle  may  be  inscribed  in,  any  regular 
polygon. 


Wesleyan  University,  September,  1896. 

li  HOURS. 

1.  The  exterior  angles  of  a  polygon,  made  by  producing  each 
of  its  sides  in  succession,  are  together  equal  to  four  right 
angles. 

The  sum  of  the  interior  angles  of  a  polygon  is  ten  right 
angles.    How  many  sides  has  the  polygon  ? 

2.  An  angle  inscribed  in  a  circle  is  measured  by  one-half  of 
the  arc  intercepted  between  its  sides. 

3.  Show  how  to  bisect  a  given  angle. 

4.  The  radius  of  a  circle  is  6  feet.  What  are  the  radii  of  the 
circles  concentric  with  it  whose  circumferences  divide  its  area 
into  three  equivalent  parts  ? 

5.  Show  how  to  inscribe  in  a  given  circle  a  regular  polygon 
similar  to  a  given  regular  polygon. 

6.  If  two  polygons  are  composed  of  the  same  number  of 
triangles,  similar  each  to  each,  and  similarly  placed,  the  poly- 
gons are  similar. 


90  GEOMETRY— NUMERICAL  PROBLEMS, 

The  University  of  Chicago,  September,  1896. 

TIME  AliLOWED,    ONE   HOUR  AND  FIFTEEN  MINUTES. 

[When  required,  give  all  reasons  in  full,  and  work  out  proofs  and  problems  in 

detail.] 

1.  Show  that  if  on  a  diagonal  of  a  parallelogram  two  points 
be  taken  equally  distant  from  the  extremities,  and  these  points 
be  joined  to  the  opposite  vertices  of  the  parallelogram,  the 
four-sided  figure  thus  formed  will  be  a  parallelogram. 

2.  State  and  prove  the  converse  of  the  following  theorem  : 
In  the  same  circle,  equal  chords  are  equally  distant  from 

the  centre. 

3.  Given  a  circle,  a  point,  and  two  straight  lines  meeting  in 
the  point  and  terminating  in  the  circumference  of  the  circle. 
State  what  four  lines  or  segments  form  a  proportion  and  in 
what  order  they  must  be  taken  : 

(1)  When  the  point  is  outside  the  circle,  and 
{a)  both  lines  are  secants, 

(&)  one  line  is  a  secant,  and  the  other  a  tangent, 
(c)  both  lines  are  tangents. 
(3)  When  the  point  is  within  the  circle,  and  the  two  lines 
are  chords. 

Prove  in  full  (1)  {a).  Show  that  (1)  (c)  is  a  limiting  case  of 
(1)  (a). 

4.  To  a  given  circle  draw  a  tangent  that  shall  be  perpendic- 
ular to  a  given  line. 

5.  Show  how  to  construct  a  triangle,  having  given  the  base, 
the  angle  at  the  opposite  vertex,  and  the  median  from  that 
vertex  to  the  base.  Discuss  the  cases  depending  upon  the 
length  of  the  given  median. 


GEOMETRY— NUMERICAL  PROBLEMS.  91 

Massachusetts  Institute  of  Technology,  June,  1896. 

[Every  reason  must  be  stated  in  full.] 

1.  If  straight  lines  are  drawn  to  the  extremities  of  a  straight 
line  from  any  point  in  the  perpendicular  erected  at  its  middle 
point,  they  make  equal  angles  with  the  line  and  with  the  per- 
pendicular. 

3.  Two  right  triangles  are  equal  when  the  hypotenuse  and 
a  side  of  one  are  equal,  respectively,  to  the  hypotenuse  and  a 
side  of  the  other. 

3.  Prove  the  formula  for  the  sum  of  the  angles  of  any  poly- 
gon. Define  a  regular  polygon.  How  many  degrees  in  each 
angle  of  a  regular  heptagon  ? 

4.  In  the  same  circle  or  in  equal  circles  chords  equally  dis- 
tant from  the  centre  are  equal. 

5.  Two  triangles  are  similar  when  their  homologous  sides 
are  proportional. 

6.  A  hexagon  is  formed  by  joining  in  succession  the  middle 
points  of  the  sides  of  a  given  regular  hexagon.  Find  the  ratio 
of  the  areas  of  these  two  hexagons. 

7.  If  A  B  and  A  i  B  i  are  any  two  chords  of  the  outer  of  two 
concentric  circles,  which  intersect  the  circumference  of  the 
inner  circle  at  P,  Q,  and  Pi,  Q i,  respectively,  prove  : 
AP.  PB=AiPx.  PiBi. 


92  OEOMETBY—NUMEBIOAL  PR  WLEMS 

Brown  University,  June,  1896. 

1.  Have  you  been  over  all  the  required  work  ? 

2.  The  exterior  angle  of  a  triangle  is  equal  to  the  sum  of 
the  opposite  interior  angles. 

3.  Find  a  point  equidistant  from  two  given  points  P  and 
Q,  and  at  a  given  distance  C  D  from  a  given  line  A  B. 

4.  If  a  secant  and  a  tangent  be  drawn  from  a  point  without 
a  circle,  the  tangent  is  a  mean  proportional  between  the  secant 
and  its  external  segment. 

5.  Similar  triangles  are  to  each  other  as  the  squares  of  their 
homologous  sides. 

6.  The  diagonals  drawn  from  a  vertex  of  a  regular  penta- 
gon to  the  opposite  vertices  trisect  that  angle. 


Vassar  College,  September,  1895. 

1.  Find  the  area  of  a  right  triangle  if  the  perimeter  is  60 
feet,  and  its  sides  are  as  3  :  4  :  5. 

2.  The  sides  of  a  triangle  are  8,  9,  13  ;  is  the  greatest  an^le 
acute,  right,  or  obtuse  ? 

3.  The  perpendicular  erected  at  the  middle  point  of  the  base 
of  an  isosceles  triangle  passes  through  the  vertex  and  bisects 
the  angle  at  the  vertex. 

4.  If  two  circles  touch  internally,  and  the  diameter  of  the 
smaller  is  equal  to  the  radius  of  the  larger,  the  circumference 
of  the  smaller  bisects  every  chord  of  the  larger  which  can  be 
drawn  through  the  point  of  contact. 

5.  If  two  similar  triangles  A  B  C,  D  E  F,  have  their  homo- 
logous sides  parallel,  the  lines  AD,  BE,  C  F  which  join  their 
homologous  vertices  meet  in  the  same  point. 


GEOMETRY— NUMERICAL  PROBLEMS.  93 


Vassar  College,  June,  1896. 

1.  Define  similar  triangles. 

State  all  the  cases  of  similar  triangles,  and  prove  one. 

2.  Construct  a  right  triangle,  having  given  the  hypotenuse 
and  the  sum  of  the  legs. 

3.  Prove  that  the  radius  of  a  circle  inscribed  in  an  equi- 
lateral triangle  is  equal  to  one-third  of  the  altitude  of  the  tri- 
angle. 

4.  Construct  the  fourth  proportional  when  three  are  given. 

5.  Find  the  area  of  an  isosceles  triangle  if  the  base  is  equal 
to  36  feet  and  one  leg  is  equal  to  30  feet. 

6.  To  divide  a  given  line  in  extreme  and  mean  ratio.  What 
regular  inscribed  polygons  may  be  constructed  by  means  of 
this  division  ?    Prove  your  statement. 


Amherst  College,  June,  1895. 

1.  To  construct  a  square  that  shall  have  to  a  given  square 
the  ratio  of  3  to  2. 

2.  The  circumference  of  a  circle  is  the  limit  of  the  perimeter 
of  a  regular  circumscribed  polygon,  as  the  number  of  sides  of 
the  polygon  is  indefinitely  increased. 

3.  If  two  polygons  are  composed  of  the  same  number  of 
similar  triangles,  similarly  placed,  the  polygons  are  similar. 

4.  The  sum  of  the  squares  on  two  sides  of  a  triangle  is  equal 
to  twice  the  square  on  half  the  third  side  increased  by  twice 
the  square  on  the  median  to  that  side. 

5.  Find  the  locus  of  all  points,  the  perpendicular  distances 
of  which  from  two  intersecting  lines  are  to  each  other  as  3 
to  2. 


94  OEOMETBY— NUMERICAL  PROBLEMS. 


Amherst  College,  June,  1896. 

1.  Two  triangles  having  an  angle  of  the  one  equal  to  an  angle 
of  the  other,  and  the  including  sides  proportional  are  similar. 

2.  Inscribe  a  circle  in  a  given  triangle. 

3.  (1)  When  are  two  lines  said  to  be  ineommensurahle  f 
(2).  Are  3f  and  Syy  incommensurable  ?  Give  the  reason  for 
your  answer.  (3).  Define  a  limit  Mention  some  propositions 
to  which  the  method  of  limits  is  applied. 

4.  In  an  isosceles  right  triangle  either  leg  is  a  mean  propor- 
tional between  the  hypotenuse  and  the  perpendicular  upon  it 
from  the  vertex  of  the  right  angle. 

5.  The  area  of  an  inscribed  regular  hexagon  is  equal  to  f  of 
that  of  the  circumscribed  regular  hexagon. 


Dartmouth  College,  1894. 

1.  Name  the  different  classes  of  triangles. 

2.  What  are  the  conditions  of  similarity  in  triangles  ? 

3.  The  diameter  of  a  circle  is  25  feet.  What  is  the  perpen- 
dicular distance  to  the  circumference  from  a  point  in  the  dia- 
meter 5  feet  from  either  end. 

4.  One  angle  of  a  parallelogram  is  |  of  a  right  angle.  What 
values  have  the  remaining  angles  ? 

5.  The  segments  of  a  given  line  are  4,  6,  7.  Divide  any 
other  line  in  the  same  proportion. 

6.  In  any  triangle  the  product  of  any  two  sides  is  equal  to 
the  product  of  the  segments  of  the  third  side  formed  by  the 
bisector  of  the  opposite  angle,  plus  the  square  of  the  bisector. 
Demonstrate. 


GEOMETRY— NUMERICAL  PROBLEMS.  95 


Wellesley  College,  June,  1896. 

1.  An  angle  formed  by  two  tangents  is  how  measured? 
Prove. 

2.  The  diagonals  of  a  rhombus  bisect  each  other  at  right 
angles. 

3.  (a)  If  a  line  bisects  an  angle  of  a  triangle  and  also  bisects 
the  opposite  side  the  triangle  is  isosceles. 

(&)  State  and  demonstrate  the  general  case  for  the  ratio  of 
the  segments  of  the  side  opposite  to  a  bisected  angle. 

4.  With  a  given  line  as  a  chord,  construct  a  circle  so  that 
this  chord  shall  subtend  a  given  inscribed  angle. 

5.  (a)  On  a  circle  of  4  feet  radius,  how  long  is  an  arc  in- 
cluded between  two  radii  forming  an  angle  of  30°  ?  Prove,  de- 
riving the  formula  employed. 

(&)  Find  the  area  of  the  regular  circumscribed  hexagon  of 
a  circle  whose  radius  is  1. 

6.  Two  similar  triangles  are  to  each  other  as  the  squares  of 
their  homologous  sides. 


Bowdoin  College,  June,  1895. 

1.  The  perpendiculars  from  the  vertices  of  a  triangle  to  the 
opposite  sides  meet  in  a  common  point. 

3.  Upon  a  given  straight  line  describe  an  arc  of  a  circle 
which  shall  contain  a  given  angle. 

3.  In  any  triangle  the  square  of  a  side  opposite  an  acute 
angle  equals  the  sum  of  the  squares  on  the  other  two  sides 
minus  twice  the  product  of  one  of  these  sides  by  the  projec- 
tion of  the  other  upon  it. 


96  GEOMETRY— NUMERICAL  PROBLEMS. 

4.  The  length  of  a  tangent  to  a  circle,  from  a  point  eight 
units  distant  from  the  nearest  point  on  the  circumference,  is 
twelve  units.     Find  the  diameter  of  the  circle. 

5.  Two  triangles  having  an  angle  of  one  equal  to  an  angle 
of  the  other  are  to  each  other  as  the  product  of  the  sides  in- 
cluding the  equal  angles. 

6.  Find  the  ratio  of  the  radius  of  a  circle  to  the  side  of  the 
inscribed  square. 

7.  The  area  of  a  sector  of  sixty  degrees  is  two  hundred  nine, 
and  forty-four  hundredths  square  inches.  Find  the  length 
of  the  radius. 


Bowdoin  College,  June,  1896. 

1.  The  bisectors  of  the  three  angles  of  a  triangle  meet  in  the 
centre  of  the  inscribed  circle. 

2.  The  circumference  of  a  circle  described  on  one  of  the  equal 
sides  of  an  isosceles  triangle  as  a  diameter  passes  through  the 
middle  point  of  the  base. 

3.  If  two  chords  be  drawn  through  a  fixed  point  within  a 
circle,  the  product  of  the  segments  of  one  chord  equals  the 
product  of  the  segments  of  the  other. 

4.  The  radius  of  a  circle  is  10  ;  inscribe  within  it  a  regular 
decagon  and  compute  the  length  of  its  side. 

5.  In  an  acute-angled  triangle  the  side  AB  =  10,  side 
A  C  =  7,  the  projection  of  A  C  on  A  B  is  3.4.  Construct  the 
triangle  and  compute  the  third  side,  B  C. 

6.  The  area  of  one  circle  is  100 ;  find  the  circumference  of 
another  circle  described  on  the  radius  of  the  first  as  a  di- 
ameter. 


GEOMETRY— NUMEBIGAL  PROBLEMS,  97 


TTniversity  of  California,  August,  1896. 

1.  Prove  that  if  two  sides  of  one  of  two  triangles  be  equal  to 
two  sides  of  the  other,  and  the  angles  opposite  one  pair  of 
equal  sides  be  equal,  the  angles  opposite  the  other  pair  of  sides 
are  either  equal  or  supplementary. 

2.  To  construct  a  triangle  having  given  the  base,  one  angle 
at  the  base  and  the  altitude. 

3.  Prove  that  the  straight  lines  drawn  at  right  angles  to  the 
sides  of  a  triangle  at  their  middle  points  meet  in  a  point. 

4.  Prove  that,  if  an  angle  at  the  centre  of  a  circle  and  an 
angle  at  the  circumference  be  subtended  by  the  same  arc,  the 
angle  at  the  circumference  is  one-half  of  the  angle  at  the 
centre. 

5.  If  the  middle  points  of  adjacent  sides  of  a  convex  quadri- 
lateral be  connected  by  straight  lines  what  figure  is  formed  ? 
What  is  the  relation  between  the  areas  of  this  figure  and  the 
quadrilateral  ?    Prove  your  statements. 

6.  To  divide  a  given  straight  line  internally  in  extreme  and 
mean  ratio.  What  regular  polygons  may  be  inscribed  in  a 
circle  by  means  of  this  construction  ?  Show  (without  proof) 
how  one  of  these  polygons  is  constructed. 

7.  Present,  in  the  clearest  language  and  most  perfect  form 
you  can  command,  some  proposition  of  your  own  choosing. 


98  OEOMETRT—NUMEBIGAL  PROBLEMS. 

Bryn  Mawr  College,  September,  1896. 

TWO  AND   ONE-HALF  HOURS. 

1.  Show  how  to  draw  a  perpendicular  from  a  given  point  to 
a  given  line,  the  point  not  lying  on  the  line.  Show  that  only 
one  such  perpendicular  can  be  drawn. 

2.  Prove  that  if  two  parallel  lines  are  cut  by  a  third  straight 
line,  the  two  interior  angles  on  one  side  of  the  transversal  are 
together  equal  to  two  right  angles. 

Prove  that  the  lines  bisecting  the  angles  of  a  parallelogram 
form  a  rectangle. 

3.  Define  a  parallelogram  ;  prove  that  the  opposite  sides  and 
angles  are  equal,  and  that  the  diagonals  bisect  one  another. 

Prove  that  any  line  through  the  intersection  of  the  diagonals 
of  a  parallelogram  bisects  the  figure. 

4.  Prove  that  in  any  circle  angles  at  the  centre  have  the 
same  ratio  as  the  arcs  on  which  they  stand. 

Show  how  to  divide  the  circumference  of  a  circle  into  three 
parts  that  shall  be  in  the  ratio  1:2:3. 

5.  Prove  that  an  angle  formed  by  two  chords  intersecting 
within  a  circle  is  measured  by  one-half  the  sum  of  the  inter- 
cepted arcs. 

A  B  C  D  is  a  quadrilateral  in  a  circle  ;  P,  Q,  R,  S,  are  the 
points  of  bisection  of  the  arcs  A  B,  B  C,  CD,  DA.  Show  that 
P  R  is  perpendicular  to  Q  S. 

6.  Prove  that  the  sum  of  the  squares  of  two  sides  of  a  tri- 
angle is  equal  to  twice  the  square  of  half  the  base  increased  by 
twice  the  square  of  the  distance  from  the  vertex  to  the  bisec- 
tion of  the  base.  Apply  this  to  find  a  line  whose  extremities 
shall  lie  one  on  each  of  two  given  concentric  circles,  the  line 
itself  being  bisected  at  a  given  point. 

7.  Prove  that  three  lines  drawn  through  the  vertices  of  a 
triangle  to  bisect  the  opposite  sides  meet  in  a  point,  and  de- 


GEOMETUT— NUMERICAL  PROBLEMS.  99 

termine  the  position  of  tliis  point  on  any  one  of  the  three 
bisectors. 

Show  how  to  construct  a  triangle  when  the  lengths  of  the 
three  medians  are  given. 

8.  Define  the  tangent  to  a  circle  at  a  point ;  and  prove  that 
the  tangent  at  a  point  is  perpendicular  to  the  diameter  through 
the  point. 

Two  circles  whose  centres  are  A,  B,  meet  at  a  point  P.  Prove 
that  if  A  P  touch  the  circle  whose  centre  is  B,  then  B  P  will 
touch  the  circle  whose  centre  is  A. 

9.  State  and  prove  the  relation  between  the  segments  of  in- 
tersecting chords  of  a  circle.  Apply  this  to  find  a  mean  pro- 
portional to  two  given  lines. 


Boston  University,  June,  1896. 

TIME    1    H.    30  M. 
[Candidates  will  quote  authority  for  each  step.] 

1.  The  extremities  of  the  base  of  an  isosceles  triangle  are 
equally  distant  from  the  opposite  sides.     Prove. 

2.  Two  unequal  circles  have  a  common  centre.  Prove  that 
chords  of  the  greater  circle,  which  are  tangent  to  the  lesser 
circle,  are  equal. 

3.  The  sides  of  a  triangle  are  4,  7,  10  ;  find  the  sides  of  a 
similar  triangle  having  nine  times  the  area  of  the  first.  Prove 
the  principle  employed. 

4.  Homologous  altitudes  of  similar  triangles  have  the  same 
ratio  as  any  two  homologous  sides.     Prove. 

5.  The  sum  of  the  perpendiculars  from  any  point  within  an 
equilateral  triangle  to  the  three  sides  is  equal  to  the  altitude 
of  the  triangle.     Prove. 


100  GEOMETRY— NUMERICAL  PROBLEMS, 

Boston  University,  September,  1896. 

TIME   1    H.    30  M. 
[Candidates  will  quote  authority  for  each  step.] 

1 .  Connect  the  mid  points  of  the  adjacent  sides  of  a  rhom- 
bus and  prove  character  of  the  figure  formed. 

2.  Chords  meeting  a  diameter  at  the  same  point  and  making 
the  same  angle  with  it  are  equal.     Prove. 

3.  The  radius  of  a  circle  is  10  feet.  Find  the  side  of  an 
equilateral  triangle  having  the  same  area  as  the  circle. 

4.  In  any  triangle  the  square  of  the  side  opposite  an  acute 
angle  is  equal  to  the  sum  of  the  squares  of  the  other  two  sides 
diminished  by  twice  the  product  of  one  of  these  sides  and  the 
projection  of  the  other  side  upon  it.     Prove. 

5.  Two  equivalent  triangles  have  a  common  base  and  lie  on 
opposite  sides  of  it.  Prove  that  the  line  joining  their  vertices 
is  bisected  by  the  base,  produced,  if  necessary. 


Vanderbilt  University,  May  24,  1894. 

1.  The  circles  described  on  two  sides  of  a  triangle  as  diame- 
ters intersect  on  the  third  side. 

2.  The  diagonals  of  a  trapezoid  divide  each  other  into  seg- 
ments which  are  proportional. 

3.  Similar  triangles  are  as  the  squares  of  their  homologous 
sides. 

4.  Two  quadrilaterals  are  equivalent  when  the  diagonals  of 
one  are  respectively  equal  and  parallel  to  the  diagonals  of  the 
other. 

5.  The  area  of  a  ring  bounded  by  two  concentric  circumfer- 
ences is  equal  to  the  area  of  a  circle  having  for  its  diameter 


OEOMETRT— NUMERICAL  PROBLEMS.  101 

a  chord  of  the  outer  circumference  tangent  to  the  inner  cir- 
cumference. 

6.  A  swimmer  whose  eye  is  at  the  surface  of  the  water  can 
just  see  the  top  of  a  stake  a  mile  distant ;  the  stake  proves  to 
be  eight  inches  out  of  the  water ;  required  the  radius  of  the 
earth. 


New  Jersey  State  College  for  the  Benefit  of  Agriculture 
and  the  Mechanic  Arts,  New  Brunswick,  N.  J.,  June, 
1891. 

1.  Define  the  various  kinds  of  triangles  and  quadrilaterals. 

2.  If  two  straight  lines  cut  each  other,  the  vertical  angles 
are  equal. 

3.  An  angle  formed  by  a  tangent  and  a  chord  from  the  point 
of  contact  is  measured  by  one-half  the  intercepted  arc. 

4.  If  a  variable  tangent  meets  two  parallel  tangents  it  sub- 
tends a  right  angle  at  the  centre. 

5.  The  bisector  of  an  angle  of  a  triangle  divides  the  opposite 
side  into  segments  proportional  to  the  adjacent  sides. 

6.  A  parallelogram  is  divided  by  its  diagonals  into  four  tri- 
angles of  equal  area. 

7.  The  areas  of  two  similar  segments  are  to  each  other  as 
the  squares  of  their  radii. 

8.  The  diameter  of  a  circle  is  5  feet ;  find  the  side  of  the  in- 
scribed square. 

9.  Find  a  side  of  the  circumscribed  equilateral  triangle,  the 
radius  of  the  circle  being  1/3, 

10.  Find  the  radius  of  the  circle  in  which  the  sector  of  45° 
is  .125  square  inches. 


LOGAEITHMS. 

1 .  The  logarithm  of  a  number  is  the  exponent  of  the 
power  to  which  an  assumed  number  must  be  raised  to  pro- 
duce the  first  number. 

2.  Since  logarithms  are  exponents,  the  principles  estab- 
lished in  Theory  of  Exponents  in  Algebra,  hold  in  loga- 
rithms, and  are  the  very  principles  which  make  logarithms 
serviceable  ;  as  follows  : 

I.  The  logarithm  of  a  product  is  equal  to  the  sum  of  the 
logarithms  of  its  factors. 

II.  The  logarithm  of  a  quotient  is  equal  to  the  logarithm  of 
the  dividend  minus  the  logarithm  of  the  divisor. 

III.  The  logarithm  of  any  power  of  a  number  is  equal  to 
the  logarithm  of  the  number  multiplied  by  the  exponent  of  the 
power. 

IV.  The  logarithm  of  a  root  of  a  number  is  equal  to  the 
logarithm  of  the  number  divided  by  the  index  of  the  root. 

3.  The  only  kind  of  logarithms  with  which  we  have  to 
do  here  are  those  in  which  the  assumed  number,  called  the 
base,  is  10. 

Such  logarithms  are  termed  Common  Logarithms. 

1 


10^  =  10000 

i«-'  =  io^  = 

.1 

103=  1000 

^'-'  =  w  = 

.01 

102  ^   100 

10- =  103  = 

.001 

101  ^   10 

^'-'=w= 

.0001 

100=    1 

LOGARITHMS.  103 

Thus,  by  definition,  log  10000  =  4  ;  log  1000  =  3,  etc. 

But  all  numbers  which  are  not  integral  powers  of  10,  as 
the  above  are,  must  have  a  fractional,  decimal,  part  to 
their  logarithms. 

Thus,  the  logarithm  of  any  number  between 
10  and  100 
would  lie  between 

that  is,  it  would  be 


1  and  2, 


1  +  a  decimal. 
Of  any  number  between 

1  and  10, 
the  logarithm  would  be 

0  +  a  decimal ; 
between  .1  and  1, 

—  1  4-  a  decimal; 

between  .01  and  .1, 

—  2  +  a  decimal ; 
and  so  on. 

This  decimal  part  of  a  logarithm  is  called  the  mantissa ; 
the  integral  part,  the  characteristic. 

From  the  above  it  is  seen  that  all  mantissas  are  positive. 
And  to  show  that  a  negative  sign  belongs  to  the  character- 
istic only,  it  is  placed  above  the  characteristic,  thus  : 

log  .03152  =  2.49859. 

4.  Moving  the  decimal  point  to  right  or  left  in  any 
number  multiplies  or  divides  that  number  by  ten  or  some 
integral  power  of  ten.  And  as  the  logarithm  of  a  product 
is  equal  to  the  logarithm  of  the  multiplicand  plus  the 


104  LOGARITHMS. 

logarithm  of  the  multiplier,  and  the  logarithm  of  a  quotient 
is  equal  to  the  logarithm  of  the  dividend  minus  the  loga- 
rithm of  the  divisor,  and  the  logarithm  of  the  multiplier 
and  divisor  in  such  cases  (moving  the  decimal  point)  is  an 
integer,  the  only  part  of  a  logarithm  affected  by  a  change 
of  the  decimal  point  in  a  number  is  the  integral  part,  the 
characteristic. 

Then  all  numbers  which  differ  only  in  the  position  of  the 
decimal  point  have  the  same  mantissa. 

5.  A  careful  study  of  Art.  3  will  make  plain  the  follow- 
ing rules  in  regard  to  the  characteristic  : 

I.  If  the  number  is  greater  than  1,  the  characteristic  is 
one  less  than  the  number  of  places  to  the  left  of  the  decimal 
point, 

II.  If  the  number  is  less  than  1,  the  characteristic  is  nega- 
tive, and  is  one  more  than  the  number  of  zeros  between  the  deci- 
mal point  and  the  first  significant  figure  of  the  decimal. 


Thus,  the  characteristic  of  log  378.37  is 

^; 

ii               i(              a     c(     ^0917  ''  — 

■3; 

t(                   (c                  a      a      5  391  ii 

0; 

((             <i            a    i(      8395  ''  — 

-1. 

6.  The  rules  for  determining  the  position  of  the  decimal 
point  in  a  number  corresponding  to  any  given  logarithm 
are  just  the  converse  of  the  above. 

I.  When  the  characteristic  is  positive,  the  number  of  places 
to  the  left  of  the  decimal  point  is  one  more  than  the  number 
of  units  in  the  characteristic. 

II.  When  the  characteristic  is  negative,  the  number  is  a 
decimal,  and  the  number  of  zeros  between  the  decimal  point 
and  the  first  significant  figure  is  one  less  than  the  number  of 
units  in  the  characteristic. 


LOGARITHMS.  105 

7.  To  avoid  certain  difficulties  in  the  use  of  logarithms 
every  logarithm  whicli  has  a  negative  characteristic  should 
have  10.00000  —  10  (equal  to  0)  added  to  it. 

Thus,  2'.37931  should  be  written  8.37931  —  10  ; 
4.92012  ''  "  ''  6.92012  —  10 ; 
1.72082       ''       ''        "       9.72082  —  10. 

8.  The  cologarithm  of  a  number,  or  ths  arithmetical 
complement  of  the  logarithm  of  the  number,  is  the  loga- 
rithm of  the  reciprocal  of  that  number. 

Thus,  colog  317  =  log  ^\^  ;  but  by  II.,  Art.  2,  log  -^ij 
=  log  1  —  log  317  =  0  —  log  317  =  —  log  317. 

And  so  the  cologarithm  of  any  number  is  equal  to  the 
negative  logarithm  of  that  number. 

9.  Since  to  subtract  a  quantity  is  to  add  that  quantity 
with  its  sign  changed,  rule  II.,  Art.  2,  may  be  stated  : 

The  logarithm  of  a  quotient  is  equal  to  the  logarithm  of  the 
dividend  plus  the  negative  logarithm,  or  cologarithm,  of  the 
divisor. 

This  is  the  form  of  the  rule  that  should  be  invariably 
applied  in  practice. 

10.  Negative  logarithms  should  always  have  zero  in  the 
form  10.00000  — 10  added  to  them  before  they  are  em- 
ployed otherwise  in  an  example.  This  altered  form  of  the 
negative  logarithm  may  well  be  distinguished  by  the  name 
cologarithm,  and  is  so  distinguished  hereafter. 

Thus, 

1777   ( log  1777  +  (—  log  8943)  =  log  1777  +  colog  8943 

^Ogoo7o  =  1  II  II  II  II 

(  3.24969  +  (-  3.95148)  =  3.24969  +  6.04851  -  10 

9.39820  -  10. 


i= 


106  LOGARITHMS. 

11.  This  method  of  using  logarithms  avoids  all  subtrac- 
tion of  logarithms,  except  in  finding  cologarithms  ;  and 
these  are  very  easily  found  by  the  following  rule  : 

Begin  with  the  characteristic  of  the  logarithm  and  subtract 
each  figure  from  9,  except  the  last  significant  figure,  and  sub- 
tract that  from  10. 

Thus,  log  8409  =  3.92474  ;  and  colog  8409  = 
10  —  3.92474—  10  --=  6.07526  —  10. 
By  subtracting  from  left  to  right  in  this  way  the  colog- 
arithm  of  any  number  of  four  figures  or  less  can  be  read 
right  from  the  table  almost  as  easily  as  the  logarithm  it- 
self, after  some  practice. 

12.  The  following  points  should  be  carefully  noted  in 
using  logarithms  that  have  negative  characteristics  : 

1.  In  getting  the  cologarithm,  the  10  following  the 
mantissa  destroys  the  second  10  of  the  10.00000  —  10 
added. 

Thus, 

-  (9.85920  —  10)=  —  9.85920  +  10  ) 

colog  .7231  =  -J      10.00000  —  10  =     10.00000  —  10  [ 


0.14080.  ; 

2.  In  adding  or  multiplying,  superfluous  tens  should 
be  dropped. 

Thus:  adding,    9.87349  —  10 
8.96454  —  10 
18.83803  —  20  =  8.83803  —  10  ; 

Multiplying,    9.76604  —  10 

3 

29.29812  —  30  =  9.29812  —  10. 

3.  In  dividing,  a  sufiicient  number  of  tens  should  be 
added,  before  and  after  the  mantissa,  to  make  the  number 
of  tens  after  the  mantissa  equal  to  the  number  of  units  in 
the  divisor. 


LOGARITHMS.  107 

Thus,  9.76155  —  10  _  29.76155  —  30  _  „  „^„.^       ,^ 
^r — —  y.y/cUO/c  —  iXj  y 

8,98304-W  ^  38.98304-40  ^  g_^^^^^  _  ^^_ 


HOW  TO   USE  THE  TABLE. 

13.  The  first  page  of  the  table  gives  the  characteristics 
and  mantissas  of  numbers  up  to  100.  The  remainder  of  the 
table  gives  only  mantissas.  The  characteristic  is  to  be  sup- 
plied by  the  rules  of  Art.  5.  The  first  three  figures  of  the 
number  are  found  in  the  left-hand  column,  marked  N  at 
the  top  and  the  bottom.  The  fourth  figure  of  the  number 
is  found  in  the  first  line  at  the  top  and  the  bottom.  The 
mantissa  is  then  found  in  the  same  horizontal  line  with  the 
first  three  figures,  and  in  the  same  vertical  column  with 
the  fourth  figure.  The  first  two  figures  of  the  mantissa 
are  printed  only  in  the  first  column.  In  every  case  where 
an  asterisk  is  found  the  first  two  figures  of  the  mantissa  are 
found  in  the  first  column  of  the  next  line  below. 

14.  To  find  the  logarithm  of  a  number. 

1.  To  find  the  logarithm  of  a  number  of  four  figures,  as 
8713. 

By  Art.  5,  the  characteristic  =  3. 

By  the  table  as  explained  above  the  mantissa  =  .94017. 

Hence  log  8713  =  3.94017. 

2.  To  find  the  logarithm  of  a  number  of  five  or  more 
figures,  as  35647. 

The  characteristic  =  4. 
The  mantissa  for  3564  =  .55194. 
''  3565  =  .55206. 
That  is,  an  increase  of  one  unit  in  the  number,  at  this 
point  in  the  table,  makes  an  increase  of  .00012  in  the  man- 


108  LOGARITHMS. 

tisga.  Then  an  increase  of .  7  of  a  unit  (7  in  the  fifth  place  is 
.  7  of  1  in  the  fourth  place)  in  the  number  will  make  an  in- 
crease of  .7  of  .00012  in  the  mantissa  =  .000084. 

(  4.55194 

Therefore  log  35647  =  ]    .00008 

(4.55202. 

Note  1. — The  difference  between  any  two  consecutive 
mantissas,  as  .00012  above,  is  called  the  tabular  difference, 
and  is  printed  in  the  right-hand  column  of  the  table  under 
D. 

Note  2. — When  all  these  tabular  differences  are  multi- 
plied by  the  nine  significant  digits  expressed  as  tenths, 
they  give  a  table  of  proportional  parts.  This  table  furnishes, 
ready-made,  the  amounts  to  be  added  to  obtain  logarithms 
of  five-figure  numbers.  Only  a  portion  (the  most  helpful, 
however)  of  such  a  table  of  proportional  parts,  is  given 
with  this  table  of  logarithms,  p.  115.  It  is  sufficient  to 
make  their  use  and  meaning  plain. 

Note  3. — In  calculating  proportional  parts  and  in  all 
calculations  with  tabular  differences  they  are  treated  as 
whole  numbers,  as  they  bear  the  same  relation  to  their 
mantissas  that  whole  numbers  do  to  whole  numbers. 

Note  4. — In  calculating  additions  to  be  made  to  a  loga- 
rithm all  figures  that  follow  the  fifth  are  rejected.  When 
the  sixth  figure  is  5,  or  greater,  the  fifth  figure  is  increased 
by  1.  When  the  last  significant  figure  of  a  logarithm  is  5, 
it  means  that  such  an  increase  has  been  made  for  rejected 
figures  following  the  fifth  place. 

3.  To  find  the  logarithm  of  18.7432. 
The  characteristic  =  1. 

[As  already  explained  in  Art.  4,  the  position  of  the  deci- 
mal point  does  not  affect  the  mantissa  in  the  least.] 


LOGARITHMS.  109 

Mantissa  for  1874  =  27274. 
"    1875  =  27300. 

That  is  an  increase  of  one  in  the  number  here  makes  an 
increase  of  26  in  the  mantissa.  Then  an  increase  of  .32  of 
one  (32  following  the  fourth  place  is  .32  of  1  in  the  fourth 
place)  in  the  number  will  make  an  increase  of  .32  of  26  in 

the  mantissa  =  8.32. 

(  1.27274 

Hence  log  18.7432  =  \ 8 

(1.27282. 

Note  5. — The  process  employed  in  finding  the  logarithm 
of  a  number  of  more  than  four  figures  is  called  interpolation. 

15.  How  to  find  the  number  corresponding  to  the  logarithm. 

1.  To  find  the  number  corresponding  to  the  logarithm 
0.56514. 

The  mantissa  increases  constantly  throughout  the  table. 
Follow  the  first  column  of  mantissas  till  56  is  found,  as 
the  first  two  figures  of  the  mantissa.  Continuing  514  is 
easily  found  in  the  same  horizontal  line  with  367  and  in 
the  column  under  4. 

Hence  the  number  (placing  the  decimal  point  by  Art. 
6)  =  3.674. 

2.  To  find  the  number  corresponding  to  the  logarithm 
8.26470  —  10. 

This  mantissa  cannot  be  found  in  the  table. 

The  nearest  mantissa  less      than  26470  =  26458. 
''        "  "       larger     ''    26470  =  26482. 

The  number  corresponding  to  mantissa  26458  (disregard- 
ing the  decimal  point)  is  1839.  For  a  mantissa  24  greater 
(26482)  the  corresponding  number  is  1840,  that  is,  an  in- 
crease of  24  in  the  mantissa,  at  this  point  in  the  table, 
means  an  increase  of  1  in  the  number.  Then  an  increase 
of  12,  which  is  the  amount  the  given  mantissa,  26470,  ex- 


110  LOGARITHMS. 

ceeds  the  mantissa  26458,  would  mean  an  increase  of  J| 
of  I,  ^  .5. 

Hence  the  number  =  .018395. 

3.  To  find  the  number  corresponding  to  the  logarithm 
1.71895. 

The  next  smaller  mantissa  =  71892. 

Then  the  given  mantissa  is  3  larger  ;  and  as  the  tabular 
difference  is  8,  |  of  1  =  .375  must  be  added  to  5235  the 
number  corresponding  to  mantissa  71892. 

Hence  the  number  =  52.3538. 

Note  1. — Numbers  corresponding  to  given  logarithms 
should  not  be  carried  to  more  than  five  or  six  significant 
figures,  in  a  five-place  table. 

Note  2. — Art.  4  makes  it  clear  that  the  mantissa  for 
200  is  the  same  as  the  mantissa  for  2000  ;  for  375,  the  same 
as  for  3750,  etc.  So  the  mantissa  for  any  number  of  three 
figures  is  found  in  the  0  column  and  in  the  same  horizontal 
line  with  these  three  figures  in  the  N  column. 

Note  3. — A  negative  quantity  cannot  be  a  power  of  a 
positive  quantity,  and  hence  a  negative  quantity,  as  such, 
has  no  logarithm.  Hence  when  negative  quantities  occur 
in  any  example  worked  by  logarithms,  the  negative  sign  is 
absolutely  disregarded,  except  so  far  as  it  affects  the  sign 
of  the  result. 


EXAMPLES. 

16.  Find  by  logarithms  the  values  of  the  following  : 

394.1  X  .9385     .    .  ^ 

1.  Given  ^  =  --^^^^3—;  find  ^. 

log  394.1  =  2.59561 
log  .9385  =  9.97243  -  10 
colog  .02003  =  1.69832 

log  X  =  4.26636  =  log  18465.4 

.-.  2:=:  18465.4 

^    ^.  (801.012) 2 X  (.0315)3. 

2.  Given  x  ^-  -^ ^^j^^, ;  find  x. 

log  (801.012)  2  =  2.90364    x  2  =  5.80728 

log   (.0315)3  =  (8.49831  -  10)  x  |  -  7.74747  -  10 

colog (1.3907)*=  (9.85677  -  10)  x  ^  =  9.97135  -  10 

loga;=  3.52610 

...  :z;  =  3358.2 

3.  95.37  X  .0313. 

4.  (-  93985)  X  1.0484. 

5.  .0008601  X  1.28865. 
5008.4 


9.394 
.93284  X  91.3009 

10.1029 
-^4 

9.8743' 


112  LOGARITHMS. 

9. 


10. 


.03494  X  (-  9432) 
.00411  X  3753.6 
-  111.121 


-  4.943 
11.  (3.1835)  ^ 

12    ^ 
(1197)§ 

13.  (.311)«. 

14.  v^.  0000009431. 
15    '      "' 


(-f)' 


16.  34985  X  (.00039)* 

F9i)i 

17.  (51^)1. 

jg    (— 419)§  X  (—90.071) 

*  (10016) «  X  (—.11101)  X  1399* 

j9     3  /(1.0642)'^x.l098 
V         (683.51)8 

^^-  v^9i7  Afnooff: 

21.       7^02053  X  .0010997  x  .32024 

^        .091352 
22  (1^)' 


23  /311  X  497  X  7.3  U 
I     ( 1 9843000)  V* 

26.  ^|-^^|. 


LOGARITHMS,  113 

27.  •|(1000)?-^(80009)U! 

28.  (911  X  10003)1. 

29      '7.40071  X  (.00352)r 

V  (.09045321)*       * 

30.  (-.l)i  X  (lOOO)i  X  ^."OT 

31.  (3+)2i 

33.  (21|)«-2-i.(80J)i-3. 

34  ^-7(444)^  X  (.00041007)^-^ 
'   V  (9.8563)i 

35  "  /  (15.434)-^  X  (3897.3)1^^  x  .41984 

V  (.000372)2-3  X  (784.96)3  x  5013.4  x  (.003)4* 


TABLES 


COMMON  LOGARITHMS  OP  NUMBERS 

Giving  Characteristics  and  Mantissas  of  Logarithms  of  Numbers 

FROM  1  to  100,  AND  MANTISSAS  ONLY  OF  NUMBERS  FROM  100  TO  10000. 


LOGARITHMS  OP  NUMBERS. 


N 

Log. 

N 

Log. 

N 

Log. 

N 

Log. 

1 

0.00000 

26 

1.41497 

51 

1.70757 

76 

1.88081 

2 

0.30103 

27 

1.43136 

52 

1.71600 

77 

1.88e>49 

3 

0.47712 

28 

1.44716 

53 

1.72428 

78 

1.89209 

4 

0.60206 

29 

1.46240 

54 

1.73239 

79 

1.89763 

5 

0.69897 

30 

1.47712 

55 

1.74036 

80 

1.90309 

6 

0.77815 

31 

1.49136 

56 

1.74819 

81 

1.90849 

7 

0.84510 

32 

1.50515 

57 

1.75587 

82 

1.91381 

8 

0.90309 

33 

1.51851 

58 

1.76343 

83 

1.91908 

9 

0.95424 

34 

1.53148 

59 

1.77085 

84 

1.92428 

10 

1.00000 

35 

1.54407 

60 

1.77815 

85 

1.92942 

11 

1.04139 

36 

1.55630 

61 

1.78533 

86 

1.93450 

12 

1.07918 

37 

1.56820 

62 

1.79239 

87 

1.93952 

13 

1.11394 

38 

1.57978 

63 

1.79934 

88 

1.94448 

14 

1.14613 

39 

1.59106 

64 

1.80618 

89 

1.94939 

15 

1.17609 

40 

1.60206 

65 

1.81291 

90 

1.95424 

16 

1.20412 

41 

1.61278 

66 

1.81954 

91 

1.95904 

17 

1.23045 

42 

1.62325 

67 

1.82607 

92 

1.96379 

18 

1.25527 

43 

1.63347 

68 

1.83251 

93 

1.96848 

19 

1.27875 

44 

1.64345 

69 

1.83885 

94 

1.97313 

20 

1.30103 

45 

1.65321 

70 

1.84510 

95 

1.97772 

21 

1.32232 

46 

1.66276 

71 

1.8.5126 

96 

1.98227 

22 

1.34242 

47 

1.67210 

72 

1.85733 

97 

1.98677 

23 

1.36173 

48 

1.68124 

73 

1.86332 

98 

1.99123 

24 

1.38021 

49 

1.69020 

74 

1.86923 

99 

1.99564 

25 

1.39794 

50 

1.69897 

75 

1.87506 

100 

2.00000 

116 


COMMON  LOGARITHMS  OF  NUMBERS. 


N 

O 

1 

2 

3 

4 

5 

6 

7 

8 

9 

D 

100 

101 
102 
103 

104 
105 
106 

107 
108 
109 

110 

111 
112 
113 

114 
115 
116 

117 
118 
119 

120 

121 
122 
123 

124 
125 
126 

127 
128 
129 

00  000 

043 

087 

130 

173 

217 

260 

303 

346 

389 

43 

432 

860 

01284 

703 

02119 

531 

938 

03  342 

743 

475 

903 
326 

745 
160 
572 

979 
383 
782 

518 
945 

368 

787 
202 
612 

*019 
423 

822 

561 
988 
410 

828 
243 
653 

*060 
463 

862 

604 

*030 

452 

870 
284 
694 

*100 
503 
902 

647 

*072 

494 

912 
325 
735 

5^141 
543 
941 

689 

*115 

536 

953 
366 

776 

*181 
583 
981 

732 
*157 

578 

995 
407 
816 

*222 

623 

*021 

775 

*199 
620 

*036 
449 

857 

*262 

663 

*060 

817 

*242 

662 

*078 
490 
898 

*302 

703 
*iOO 

43 
42 
42 

42 
41 
41 

40 
40 
40 

04139 

179 

218 

258 

297 

336 

876 

415 

45i 

493 

39 

532 
922 

05  308 

690 

06  070 
446 

819 

07188 

555 

571 
961 
346 

729 
108 

483 

856 
225 
591 

610 
999 

385 

767 
145 
521 

893 
262 

628 

650 

*038 

423 

805 
183 
558 

930 
298 
664 

689 

*077 
461 

843 
221 
595 

967 
335 
700 

727 

*115 

500 

881 
258 
633 

3^004 

372 
737 

766 
*154 

538 

918 
296 
670 

*041 
408 
773 

805 

*192 

576 

956 
333 

707 

*078 
445 
809 

844 

*231 

614 

994 
371 
744 

*115 

482 
846 

883 

*269 

652 

*032 
408 

781 

*151 

518 
882 

39 
39 
38 

38 
38 
37 

87 

37 
36 

918 

954 

990  *027 

*063 

*099  *135 

*171 

*207 

*243 

36 

08  279 
636 
991 

09  342 
691 

10  037 

380 

721 

11059 

314 

672 

*026 

377 
726 
072 

415 
755 

093 

350 

707 
*061 

412 
760 
106 

449 
789 
126 

386 

743 

*096 

447 
795 
140 

483 
823 
160 

422 

778 
*132 

482 
830 
175 

517 
857 
193 

458 

814 

*167 

517 
864 
209 

551 

890 
227 

493 

849 
j*202 

i  552 

899 
243 

585 
924 
261 

529 

884 

*237 

587 
934 

278 

619 
958 
294 

565 

920 

*272 

621 

968 
312 

653 
992 
327 

600 

955 

*307 

656 

*003 

346 

687. 
*025 
361 

36 
35 

35 

35 

35 
34 

34 
34 
34 

N 

O 

1 

2 

3 

4 

5 

6 

7 

8 

9 

D 

PP 

1 
2 
3 

4 
5 

6 

7 
8 
9 

44 

4.4 

8.8 
13.2 

17.6 
22.0 
26.4 

30.8 
35.2 
39.6 

43 

4.3 

8.6 

12.9 

17.2 
21.5 

25.8 

30.1 
34.4 
38.7 

42 

4.2 

8.4 
12.6 

16.8 
21.0 
25.2 

29.4 
33.6 

37.8 

41 

4.1 

8.2 

12.3 

16.4 
20.5 
24.6 

28.7 
32.8 
36.9 

40 

4.0 

8.0 

12.0 

16.0 
20.0 
24.0 

28.0 
32.0 
36.0 

39 

3.9 

7.8 

11.7 

15.6 
19.5 
23.4 

27.3 
31.2 
35.1 

38 

3.8 

7.6 

11.4 

15.2 
19.0 
22.8 

26.6 
30.4 
34.2 

37 

3.7 

7.4 

11.1 

14.8 

18.5 
22.2 

25.9 
29.6 
33.3 

86 

3.6 

7.2 
10.8 

14.4 
18.0 
21.6 

25.2 
28.8 
82.4 

COMMON  LOGARITHMS  OF  NUMBERS. 


117 


N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

D 

130 

131 
182 
133 

134 
135 
136 

137 
138 
139 

140 

141 
143 
143 

144 
145 
146 

147 
148 
149 

150 

151 
153 
153 

154 
155 
156 

157 
158 
159 

11394 

438 

461 

494 

538 

561 

594 

638 

661 

694 

33 

737 

12  057 
385 

710 

13  033 
354 

672 
988 

14  301 

760 
090 
418 

743 
066 
386 

704 

*019 

333 

793 
123 
450 

775 
098 
418 

735 

*051 

364 

826 
156 
483 

808 
130 
450 

767 

*083 

395 

860 
189 
516 

840 
163 

481 

799 

*114 

436 

893 
333 

548 

873 
194 
513 

830 
*145 

457 

936 
354 
581 

905 
336 
545 

863 

*176 

489 

959 
287 
613 

937 

358 

577 

893 

*208 

530 

992 
330 
646 

969 
390 
609 

935 

*339 

551 

*024 
352 
678 

*001 
323 
640 

956 

*270 
583 

33 
33 
33 

33 

33 
32 

32 
31 
31 

613 

644 

675 

706 

737 

768 

799 

829 

860 

891 

31 

932 

15  329 
534 

836 

16  137 
435 

732 

17  036 
319 

953 
259 
564 

866 
167 
465 

761 
056 
348 

983 
390 
594 

897 
197 
495 

791 

085 
377 

*014 
320 
625 

927 

227 
534 

820 
114 
406 

*045 
351 
655 

957 
256 
554 

850 
143 
435 

W6 
381 

685 

987 

386 
584 

879 
173 
464 

*106 
412 
715 

*017 
316 
613 

909 
202 
493 

*137 
442 
746 

*047 
346 
643 

938 
331 
523 

*168 
473 
776 

*077 
376 
673 

967 
260 
551 

*198 
503 
806 

*107 
406 
702 

997 
289 
580 

31 
31 
30 

30 
30 
30 

29 
29 
39 

609 

638 

667  ;  696 

725 

754 

782 

811 

840 

869 

39 

898 

18184 

469 

752 

19  033 
312 

590 

866 

20140 

936 
213 

498 

780 
061 
340 

618 
893 
167 

955 
241 
526 

808 
089 
368 

645 
921 
194 

984 
270 
554 

837 
117 
396 

673 

948 
222 

*013 

298 
583 

865 
145 
424 

700 
976 
249 

*041 
327 
611 

893 
173 
451 

728 

9^003 

276 

*070 
355 
639 

931 

201 
479 

756 

*0b0 

303 

*099 
384 
667 

949 
339 
507 

783 

*058 

3:30 

*127 
412 
696 

977 
357 
535 

811 

*085 

358 

*156 
441 

724 

*005 

385 
563 

838 

m2 

385 

29 
29 

28 

38 
28 
28 

38 
87 
37 

N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

D 

PP 

1 
2 
3 

4 
5 

6 

7 
8 
9 

35 

3.5 

7.0 

10.5 

14.0 
17.5 
21.0 

24.5 
28.0 
31.5 

34 

3.4 

6.8 

10.2 

13.6 
17.0 
20.4 

23.8 
27.2 
30.6 

33 

3.3 
6.6 
9.9 

13.2 
16.5 
19.8 

23.1 
26.4 
29.7 

32 

3.2 
6.4 
9.6 

12.8 
16.0 
19.2 

22.4 
25.6 
28.8 

31 

3.1 
6.3 
9.3 

12.4 
15.5 

18.6 

31.7 
34.8 
27.9 

80 

3.0 
6.0 
9.0 

12.0 
15.0 
18.0 

21.0 
24.0 
27.0 

39 

3.9 

5.8 
8.7 

11.6 
14.5 
17.4 

20.3 
23.2 
36.1 

28 

3.8 
5.6 

8.4 

11.2 
14.0 
16.8 

19.6 
22.4 
25.2 

37 

3.7 
5.4 
8.1 

10.8 
13.5 
16.3 

18.9 
21.6 
24.3 

118 


COMMON  LOGARITHMS  OF  NUMBERS. 


N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

D 

160 

161 
162 
163 

20  412 

439 

466 

493 

520 

548 

575 

602 

629 

656 

27 

683 

952 

21219 

710 
978 
245 

737 

*005 

272 

763 

*032 

299 

790 

*059 

325 

817 

*085 

352 

844 
*112 

378 

871 

^139 

405 

898 

*165 

431 

925 
*192 

458 

27 
27 
27 

164 
165 
166 

484 

748 

22  011 

511 

775 
037 

537 
801 
063 

564 

827 
089 

590 
854 
115 

617 
880 
141 

&43 
906 
167 

669 
932 
194 

696 

958 
220 

722 
985 
246 

26 
26 
26 

167 
168 
169 

170 

171 
172 
173 

272 
531 
789 

298 
557 

814 

324 
583 
840 

350 

608 
866 

376 
634 
891 

401 
660 
917 

427 
686 
943 

453 
712 
968 

479 

787 
994 

505 

763 

*019 

26 
26 
26 

23  045 

070 

096 

121 

147 

172 

198 

223 

249 

274 

25 

300 
553 
805 

325 

578 
830 

350 
603 

855 

376 
629 

880 

401 
654 

905 

426 
679 
930 

452 
704 
955 

477 
729 
980 

502 

754 

*005 

528 

779 

*030 

25 
25 

25 

174 
175 
176 

24  055 
304 
551 

080 
329 
576 

105 
353 
601 

130 

378 
625 

155 
403 
650 

180 
428 
674 

204 
452 
699 

229 

477 
724 

254 
502 

748 

279 

527 
773 

25 
25 
25 

177 

178 
179 

180 

181 
182 
183 

797 

25  042 

285 

822 
066 
310 

846 
091 
334 

871 
115 
358 

895 
139 
382 

920 
164 
406 

944 
188 
431 

969 
212 
455 

993 
237 
479 

*018 
261 
503 

25 

24 
24 

527 

551 

575 

600 

624 

648 

672   696 

720 

744 

24 

768 

26  007 

245 

792 
031 
269 

816 
055 
293 

840 
079 
316 

864 
102 
340 

888 
126 
364 

912 
150 

387 

935 
174 
411 

959 
198 
435 

983 

221 

458 

24 
24 
24 

184 
185 
186 

482 
717 
951 

505 
741 
975 

529 
764 

998 

553 

788 
*021 

576 

811 
*045 

600 

834 

*C68 

623 

858 
*091 

647 

881 

*114 

670 

905 
*138 

694 

928 
*161 

24 
23 
23 

187 
188 
189 

27184 
416 
646 

207 
439 
669 

231 
462 
692 

254 

485 
715 

277 
508 
738 

300 
531 
761 

323 
554 

784 

346 

577 
807 

370 
600 
830 

393 
623 

852 

23 
23 
23 

N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

D 

PP 

27 

26 

25 

24       23 

22 

1 
2 
3 

2/ 
5.' 

8.] 

2.6 
5.2 

7.8 

2.5 
5.0 
7.5 

2.4 
4.8 

7.2 

2.3 
4.6 
6.9 

2.2 
4.4 
6.6 

4 

5 
6 

10.  { 
13.{ 
16J 

10.4 
13.0 
15.6 

10.0 
12.5 
15.0 

9.6 
12.0 
14.4 

9.2 
11.5 
13.8 

8.8 
11.0 
13.2 

7 
8 
9 

18.1 
21.( 
24.( 

18.2 
20.8 
23.4 

17.5 

20.0 
22.5 

16.8 
19.2 
21.6 

16.1 

18.4 
20.7 

15.4 
17.6 
19.8 

COMMON  LOOAIilTHMS   OF  NUMBERS. 


119 


N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

D 

190 

191 
193 
193 

875 

898 

921 

944 

967 

989 

*013 

*035 

*058 

*081 

23 

28  103 
330 
556 

136 

353 

578 

149 
375 

601 

171 
398 
633 

194 
431 
646 

217 
443 
668 

340 
466 
691 

363 

488 
713 

385 
511 
735 

307 
533 

758 

23 
23 
33 

194 
195 
196 

780 

29  003 

226 

803 
026 
248 

825 
048 
270 

847 
070 
293 

870 
093 
314 

892 
115 
336 

914 
137 

358 

937 
159 

380 

959 
181 
403 

981 
203 
425 

23 
22 
22 

197 
198 
199 

200 

201 
202 
203 

447 
667 

885 

469 

688 
907 

491 
710 
929 

513 
733 
951 

535 
754 

973 

557 

776 
994 

579 

798 

*016 

601 

820 
*038 

633 

843 
*060 

645 

863 
*08l 

22 
22 
22 

30103 

125 

146 

168 

190 

211 

333 

355 

276 

298 

23 

320 
535 
750 

341 
557 

771 

363 

578 
792 

384 
600 

814 

406 
631 

835 

428 
643 
856 

449 
664 

878 

471 
685 
899 

492 
707 
920 

514 

728 
942 

82 
81 
81 

204 

205 
206 

963 

31175 

387 

984 
197 
408 

*006 
218 
429 

*027 
339 
450 

*048 
260 
471 

*0()9 
281 
492 

*091 
303 
513 

*112 
333 
534 

*133 
345 
555 

*154 

366 
576 

81 
21 
3L 

207 
208 
209 

210 

211 
212 
213 

597 

806 

32  015 

618 

827 
035 

6:39 
848 
056 

660 

869 
077 

681 
890 
098 

702 
911 
118 

733 
931 
139 

744 
953 
160 

765 
973 
181 

785 
9i)4 
201 

81 
81 
31 

222 

243 

263 

384 

305 

325 

346 

366 

387 

408 

81 

426 
634 

838 

449 
654 

858 

469 
675 
879 

490 
695 
899 

510 

715 
919 

531 
736 
940 

552 

756 
960 

578 

777 
980 

593 

797 

*001 

613 

818 

*021 

30 
20 
20 

214 
215 
216 

33  041 
344 
445 

063 
264 
465 

082 
284 
486 

103 
304 

506 

122 
335 
536 

143 
345 
546 

163 
365 
566 

183 
385 
586 

303 
405 
606 

234 
435 
636 

20 
20 
20 

217 

218 
219 

220 

221 
222 
223 

646 

846 

34  044 

666 
866 
064 

686 

885 
084 

706 
905 
104 

726 
925 
124 

746 
945 
143 

766 
965 
163 

786 
985 
183 

806 

*005 

203 

826 

*025 

223 

20 
20 
20 

243 

262 

282 

301 

331 

341 

361 

380 

400 

420 

20 

439 
635 

830 

459 
655 
850 

479 
674 
869 

498 
694 
889 

518 
713 
908 

537 

733 
928 

557 
753 
947 

577 
773 
967 

596 
792 
986 

616 
811 

*005 

20 
19 
19 

224 
225 
226 

35  025 
218 
411 

044 
238 
430 

064 
357 
449 

083 
276 

468 

103 

395 

488 

123 
315 
507 

141 
334 
536 

160 
353 
545 

180 
372 
564 

199 
392 

583 

19 
19 
19 

227 
228 
229 

603 
798 

984 

622 

813 

*003 

641 

832 

*021 

660 

851 

*040 

679 

870 

*059 

698 

889 

*078 

717 

908 

*097 

736 

927 

*116 

755 

946 
*135 

774 

965 

*154 

19 
19 
19 

N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

D 

120 


COMMON  LOGARITHMS  OF  NUMBERS. 


N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

D 

230 

231 
232 
233 

36173 

192 

211 

229 

248 

267  286 

805 

324 

342 

19 

361 
549 
736 

380 
568 
754 

399 
586 
773 

418 
605 
791 

436 
624 
810 

455 
642 

829 

474 
661 

847 

493 

680 
866 

511 
698 

884 

580 
717 
903 

19 
19 
19 

234 
235 
236 

922 

37107 

291 

940 
125 
310 

959 
144 
328 

977 
162 
346 

996 
181 
365 

*014 
199 
383 

*033 
218 
401 

*051 
236 
420 

*070 
254 
438 

*088 
273 
457 

18 
18 
18 

237 

238 
239 

240 

241 
242 
243 

475 

658 
840 

493 

676 
858 

511 
694 

876 

530 

712 

894 

548 
731 
912 

566 
749 
931 

585 

767 
949 

603 

785 
967 

621 
803 
985 

639 

822 

*008 

18 
18 
18 

38  021 

039 

057 

075 

093 

112  1  130 

148 

166 

184 

18 

202 
382 
561 

220 
399 
578 

238 
417 
596 

256 
435 
614 

274 
453 
632 

292  310 
471  489 
650  !  668 

328 
507 
686 

346 
525 

703 

364 
543 

721 

18 
18 
18 

244 
245 

246 

739 

917 

39  094 

757 

934 
111 

775 
952 
129 

792 
970 
146 

810 
987 
164 

b28 

*005 

182 

846 

*023 

199 

863 

*041 

217 

881 

*058 

235 

899 

*076 

252 

18 
18 
18 

247 
248 
249 

250 

251 
252 
253 

270 
445 
620 

287 
463 
637 

305 
480 
655 

322 

498 
672 

340 
515 
690 

358 
533 

707 

375 
550 

724 

393 
568 
742 

410 

585 
759 

428 
602 

777 

18 
18 
17 

794 

811 

829 

846 

863 

881  1  898 

915 

933 

950 

17 

967 

40  140 

312 

985 
157 
329 

*002 
175 
346 

*019 
192 
364 

*037 
209 
381 

*054  *071 
226  243 
398  1  415 

*088 
261 
432 

*106 
278 
449 

*128 
295 
466 

17 

17 
17 

254 
255 
256 

483 
654 
824 

500 
671 
841 

518 
688 
858 

535 

705 

875 

552 
722 
893 

569  586 
739  756 
909  926 

603 

773 
943 

620 
790 
960 

637 
807 
976 

17 
17 
17 

257 
258 
259 

260 

261 
262 
263 

993 

41  162 

330 

*010 
179 

347 

*027 
196 
363 

*044 
212 

380 

*061 
229 
397 

*078  *095 
246  263 
414   430 

*111 
280 
447 

^128 
296 
464 

*145 
813 
481 

•17 
17 
17 

497 

514' 

531 

547 

564 

581  1  597 

614 

631   647 

17 

664 
880 
996 

681 

847 

*012 

697 

863 

*029 

714 

880 

*045 

731 

896 

*062 

747  764 

913  i  929 

*078  *095 

780 

946 

*111 

797 

9()8 

*127 

814 

979 

*144 

17 
16 
16 

264 
265 
266 

42160 
325 
488 

177 
341 
504 

193 
357 
521 

210 
874 

537 

226 
390 
553 

243  i  259 
406  1  423 
570  !  586 

275 
489 
602 

292 
455 
619 

308 
472 
635 

16 
16 
16 

267 
268 
269 

651 
813 
975 

667 
830 
991 

684 

846 

*008 

700 

862 

*024 

716 

878 

*040 

732  i  749 

894   911 

*056  *072 

i 

765 

927 

*088 

781   797 

943   959 

n04  *120 

16 
16 
16 

N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

D 

COMMON  LOGARITHMS  OF  NUMBERS. 


121 


N 

O 

1 

2 

3 

4 

5 

6 

7 

8 

9 

D 

270 

271 

273 
273 

43136 

153 

169 

185 

201 

317 

233 

349 

265 

381 

16 

297 
457 
616 

313 
473 

633 

339 
489 
648 

345 

505 
664 

361 
531 
680 

377 

537 
696 

393 
553 

712 

409 
569 

737 

425 

584 
743 

441 

600 
759 

16 
16 
16 

274 
275 
276 

775 

933 

44  091 

791 
949 
107 

807 
965 
123 

833 
981 
138 

838 
996 
154 

854 

»013 

170 

870 

*038 

185 

886 

*044 

301 

903 

*059 

217 

917 

*075 

233 

16 
16 
16 

277 
278 
279 

280 

281 
282 
283 

248 
404 
560 

364 
420 
576 

279 
436 
593 

295 
451 

607 

311 
467 
633 

326 
483 

638 

342 

498 
654 

358 
514 
669 

373 

539 
685 

389 
545 

700 

16 
16 
16 

716 

731 

747 

762 

778 

793 

809 

834 

840 

855 

15 

871 

45  025 

179 

886 
040 
194 

903 
056 
209 

917 
071 
225 

933 
086 
340 

948 
103 
355 

963 

117 
371 

979 
133 

386 

994 
148 
301 

*010 
163 
317 

15 
15 
15 

284 

285 
286 

333 
484 
637 

347 
500 
653 

363 
515 
667 

378 

530 
683 

393 
545 
697 

408 
561 
713 

423 
576 

728 

439 
591 
743 

454 
606 

758 

469 
631 

773 

15 
15 
15 

287 

288 
289 

290 

291 
293 
293 

788 

939 

46  090 

803 
954 

105 

818 
969 
130 

834 
984 
135 

849 

*000 

150 

864 

*015 

165 

879 

*030 

180 

894 

*045 

195 

909 

*060 

310 

934 

*075 

335 

15 
15 
15 

240 

355 

370 

285 

300 

315 

330 

845 

359 

374 

15 

389 
538 

687 

404 
553 

703 

419 
568 
716 

434 
583 
731 

449 
598 
746 

464 
613 
761 

479 
627 
776 

494 
643 
790 

509 
657 

805 

523 
673 

830 

15 
15 
15 

394 

295 
296 

835 

982 

47129 

850 
997 
144 

864 

*013 

159 

879 

*026 

173 

894 

*()41 

188 

909 

*056 

203 

933 

*070 

217 

938 

*085 

232 

953 

*100 

346 

967 

*114 

361 

15 

15 
15 

297 
298 
299 

300 

301 
302 
303 

276 
493 
567 

290 
436 
583 

30§ 
451 
596 

319 
465 
611 

334 

480 
635 

349 
494 
640 

363 
509 
654 

378 
524 
669 

393 
538 
683 

407 
553 
698 

15 

15 
15 

713 

737 

741 

756 

770 

784 

799 

813 

838 

843 

14 

857 

48  001 

144 

871 
015 
159 

885 
039 
173 

900 
044 

187 

914 

058 
303 

939 
073 
316 

943 

087 
330 

958 
101 
244 

973 
116 
359 

986 
130 
273 

14 
14 
14 

304 
305 
306 

387 
430 
573 

303 
444 
586 

316 
458 
601 

330 
473 
615 

344 

487 
639 

359 
501 
643 

373 
515 

657 

387 
530 
671 

401 
544 
686 

416 

558 
700 

14 
14 

14 

307 
308 
309 

714 
855 
996 

738 

869 

*010 

743 

883 
*024 

756 

897 
*038 

770 

911 

*053 

785 

926 

*066 

799 

940 

*080 

813 

954 

*094 

837 

968 

*108 

841 

983 
*133 

14 
14 
14 

N 

O 

1 
i 

2 

3 

4 

5 

6 

7 

8 

9 

D 

122 


COMMON  LOGARITHMS  OF  NUMBERS. 


N 

0 

1 

2 

3 

4 

1 
5 

6 

7 

8 

9 

D 

310 

311 
313 
313 

49136 

150 

164 

178 

192 

206 

220 

234  1  248 

262 

14 

276 
415 
554 

290 
429 

568 

304 
443 

582 

318 
457 
596 

332 
471 
610 

346 
485 
624 

360 
499 
638 

374 
513 
651 

388  1  402 
527   541 
665   679 

14 
14 
14 

314 

315 
316 

693 
831 
969 

707 
845 
982 

721 
859 
996 

734 

872 

*010 

748 

886 

*024 

762 

900 

*037 

776 

914 

*051 

790 

927 

*065 

803  '  817 

941  ■   955 

*079  *092 

14 
14 
14 

317 

318 
319 

320 

321 
322 
323 

50  106 
243 
379 

120 

256 
393 

133 
270 
406 

147 

284 
420 

161 
297 
433 

174 
311 
447 

188 
325 
461 

202 

338 
474 

215 
352 

488 

229 
365 
501 

14 
14 

14 

515 

529 

542 

556 

569 

583 

596 

610  623 

637 

14 

651 

786 
920 

664 
799 
934 

678 
813 
947 

691 

826 
961 

705 
840 
974 

718 
853 
987 

732 

866 

*001 

745 

880 
*014 

759 
893 

*028 

772 

907 

*041 

14 
13 
13 

324 
325 
326 

51055 
188 
322 

068 
202 
335 

081 
215 
348 

095 
228 
362 

108 
242 
375 

121 
255 

388 

135 
268 
402 

148 
282 
415 

162 
295 

428 

175 
308 
441 

13 
13 
13 

327 
328 
329 

330 

331 
332 
333 

455 

587 
720 

468 
601 
733 

481 
614 
746 

495 
627 
759 

508 
640 

772 

521 
654 

786 

534 
667 
799 

548 
680 
812 

561 
693 

825 

574 

706 

838 

13 
13 
13 

851 

865 

878 

b91 

904 

917 

930 

943  1  957 

970 

13 

983 

52  114 

244 

996 

127 
257 

*009 
140 
270 

*022 
153 

284 

*035 
166 

297 

*048 
179 
310 

*061 
192 
323 

*075 
205 
336 

*088 
218 
349 

*101 
231 
362 

13 
13 
13 

334 
335 
336 

375 

504 
634 

388 
517 
647 

401 
530 
660 

414 
543 
673 

427 
556 
686 

440 
5(>9 
699 

453 

582 
711 

466 
595 
724 

479 
608 

737 

492 
621 
750 

13 
13 
13 

337 
338 
339 

340 

341 
342 
343 

763 

892 

53  020 

776 
905 
033 

789 
917 
046 

802 
930 

058 

815 
943 
071 

827 
956 
084 

840 
969 
097 

853 
982 
110 

866 
994 
122 

879 

*007 

135 

'13 
13 
13 

148 

161 

173 

186 

199 

212 

224 

237  1  250 

263 

13 

275 
403 
529 

288 
415 
542 

301 
428 
555 

314 
441 
567 

326 
453 

580 

339 
466 
593 

352 
479 

605 

364 
491 
618 

377 
504 
631 

390 
517 
643 

13 
13 
13 

344 
345 
346 

656 

782 
908 

668 
794 
920 

681 
807 
933 

694 
820 
945 

706 
832 
958 

719 
845 
970 

732 
857 
983 

744 
870 
995 

757 

882 

*008 

769 

895 
*020 

13 
13 
13 

347 
348 
349 

54  033 

158 
283 

045 
170 
295 

058 
183 
307 

070 
195 
320 

083 
208 
332 

095 
220 
345 

108 
233 
357 

120 
245 
^70 

133 

258 
382 

145 

270 
394 

13 
12 

12 

N 

0 

1 

2 

3 

4 

5 

6    7 

1 

8 

9 

D 

COMMON  LOGARITHMS  OF  NUMBERS. 


123 


N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

D 

350 

351 
352 
353 

407 

419 

432 

444 

456 

469 

481 

494 

506 

518 

12 

531 
654 

777 

543 
667 

790 

555 

679 

802 

568 
691 
814 

580 
704 

837 

593 
716 
839 

605 
728 
851 

617 
741 

864 

630 
753 

876 

642 

765 

888 

12 
12 
12 

354 
355 
356 

900 

55  033 

145 

913 
035 
157 

925 

047 
169 

937 
060 
182 

949 
073 
194 

963 

084 
306 

974 
096 
218 

986 
108 
230 

998 
131 
242 

*011 
133 
255 

12 
12 

12 

357 
358 
359 

360 

361 
363 
363 

267 
388 
509 

279 
400 
532 

291 
413 
534 

303 
435 
546 

315 
437 
558 

338 
449 
570 

340 
461 

582 

352 
473 
594 

364 
485 
606 

376 
497 
618 

12 
12 
12 

630 

643 

654 

666 

678 

691 

703 

715 

727 

739 

12 

751 
871 
991 

763 

883 
*003 

775 

895 

*015 

787 

907 

*027 

799 

919 

*038 

811 

931 

5^050 

823 

943 

*062 

835 
955 

*074 

847 

967 

*086 

859 

979 

*098 

12 
13 
12 

364 
365 
366 

56110 
239 
348 

132 
241 
360 

134 
253 
373 

146 
265 

384 

158 
277 
396 

170 
389 
407 

182 
301 
419 

194 
312 
431 

205 
334 
443 

217 
336 
455 

12 

12 
12 

367 
368 
369 

370 

371 
372 
373 

467 

585 
703 

478 
597 
714 

490 
608 
736 

502 
620 

738 

514 
633 

750 

536 
644 
761 

538 
656 
773 

549 

667 

785 

561 
679 

797 

573 
691 

808 

12 
12 

12 

820 

832 

844 

855 

867 

984 
101 

317 

879 

891 

902 

914 

926 

12 

937 
57  054 

171 

949 
066 
183 

961 
078 
194 

972 
089 
206 

996 
113 

239 

*008 
134 
341 

*019 
136 
252 

*031 
148 
364 

*043 
159 
276 

13 
12 
12 

374 
375 
376 

287 
403 
519 

299 
415 
530 

310 
426 
543 

323 
438 
553 

334 
449 
565 

345 
461 
576 

357 
473 

588 

368 
484 
600 

380 
496 
611 

392 
507 
623 

12 
12 
12 

377 
378 
379 

380 

381 
383 
383 

634 

749 
864 

646 

761 
875 

657 

773 
887 

669 

784 
898 

680 
795 
910 

693 
807 
931 

703 
818 
933 

715 
830 
944 

736 
841 
955 

738 
852 
967 

11 
11 
11 

978 

990 

*001 

*013 

137 
240 
354 

*024 

*035 

*047 

*058 

*070 

*081 

11 

58  092 
206 
320 

104 
218 
331 

115 
239 
343 

138 
252 
365 

149 
263 

377 

161 
874 

388 

172 
286 
399 

184 
297 
410 

195 

309 
422 

11 
11 
11 

384 
385 
386 

433 
546 
659 

444 
557 

670 

456 
569 

681 

467 
580 
692 

478 
591 
704 

490 
603 
715 

501 
614 
726 

513 
635 
737 

524 
636 
749 

535 
647 
760 

11 
11 
11 

387 
388 
389 

771 
883 
995 

782 

894 

*006 

794 

906 

*017 

805 

917 

*038 

816 

938 

*040 

837 

939 

»051 

838 

950 

*063 

850 

961 

*073 

861 

973 

*084 

872 

984 

*095 

11 
11 
11 

N 

0 

1 

2 

3 

4   5 

6 

7 

8 

9 

D 

124 


COMMON  LOGARITHMS  OF  NUMBERS, 


N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

D 

390 

391 
392 
393 

59106 

118 

129 

140 

151 

162 

173 

184 

195 

207 

11 

218 
329 
439 

229 
340 
450 

240 
351 
461 

251 

362 

472 

262 
373 
483 

273 
384 
494 

284 
395 
506 

295 
406 
517 

306 
417 
528 

318 
428 
539 

11 
11 
11 

394 
395 
396 

550 
660 
770 

561 
671 
780 

572 

682 
791 

583 
693 

802 

594 
704 
813 

605 
715 

824 

616 
726 
835 

627 
737 
846 

638 

748 
857 

649 

759 
868 

11 
11 
11 

397 

398 
399 

400 

401 
402 
403 

879 

988 

60  097 

890 
999 

108 

901 

*010 

119 

912 

*021 

130 

923 

*032 

141 

934 

*043 
152 

945 

*054 
163 

956 

*065 
173 

966 
*076 

184 

977 

*086 

195 

11 
11 
11 

206 

217 

228 

239 

249 

260 

271 

282 

293 

304 

11 

314 
423 
531 

325 
433 
541 

336 
444 
552 

347 
455 
563 

358 
466 
574 

369 

477 
584 

379 

487 
595 

390 
498 
6U6 

401 
509 
617 

412 
520 

627 

11 
11 
11 

404 
405 
406 

638 
746 
853 

649 
756 
863 

660 
767 

874 

670 

778 
b85 

681 
788 
895 

692 
799 
906 

703 
810 
917 

713 

821 
927 

724 
831 
938 

735 
842 
949 

11 
11 
11 

407 

408 
409 

410 

411 
412 
413 

959 
61066 

172 

970 
077 
183 

981 
087 
194 

991 
098 
204 

*002 
109 

215 

»013 
119 

225 

*023 
130 
236 

*034 
140 

247 

*045 
151 

257 

*055 
162 
268 

11 
11 
11 

278 

289 

300 

310 

321 

331 

342 

352 

363 

374 

11 

384 
490 
595 

395 
500 
606 

405 
511 
616 

416 
521 
627 

426 
532 
637 

437 
542 

648 

448 
553 
658 

458 
563 
669 

469 
574 
679 

479 

584 
690 

11 
11 
11 

414 
415 
416 

700 
805 
909 

711 
815 
920 

721 
826 
930 

731 
836 
941 

742 
847 
951 

752 

857 
962 

763 
868 
972 

773 
878 
982 

784 
888 
993 

794 

899 

*003 

10 
10 
10 

417 
418 
419 

420 

421 
422 
423 

62  014 
118 
221 

024 

128 
232 

034 

138 
242 

045 
149 
252 

055 
159 
263 

066 
170 
273 

076 
180 

284 

086 
190 
294 

097 
201 
304 

107 
211 
315 

10 
10 
10 

325 

335 

346 

356 

366 

377 

387 

397 

408 

418 

10 

428 
531 
634 

439 
542 
644 

449 
552 
655 

459 
562 
665 

469 
572 
675 

480 
583 
685 

490 
593 
696 

500 
603 
706 

511 
613 
716 

521 
624 
726 

10 
10 
10 

424 
425 
426 

737 
839 
941 

747 
849 
951 

757 
859 
961 

767 
870 
972 

778 
880 
982 

788 
890 
992 

798 

900 

*002 

808 

910 

*012 

818 

921 

*022 

829 

931 

*033 

10 
10 
10 

427 
428 
429 

63  043 
144 
246 

053 
155 
256 

063 
165 
266 

073 
175 

276 

083 
185 
286 

094 
195 
296 

104 
205 
306 

114 

215 
317 

124 
225 

327 

134 

236 
337 

10 
10 
10 

N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

D 

COMMON  LOGARITHMS  OF  NUMBERS. 


125 


N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

D 

430 

431 
432 
433 

347 

357 

367 

377 

387 

897 

407 

417 

428 

438 

10 

448 
548 
649 

458 
558 
659 

468 
568 
669 

478 
579 
679 

488 
589 
689 

498 
599 
699 

508 
609 
709 

518 
619 
719 

528  1  538 
629  1  639 
729  739 

10 
10 
10 

434 
435 
436 

749 
849 
949 

759 
859 
959 

769 
869 
969 

779 
879 
979 

789 
889 
988 

799 
899 
998 

809 

909 

*008 

819 

919 

*018 

829  839 

929  1  939 

*028  *038 

10 
10 
10 

437 
438 
439 

440 

441 
442 
443 

64  048 
147 
246 

058 
157 
256 

068 
167 
266 

078 
177 
276 

088 
187 
286 

098 
197 
296 

108 
207 
306 

118 
217 
316 

128 
227 
326 

187 

237 
335 

10 
10 
10 

345 

355 

365 

375 

385 

395 

404 

414 

424  !  434 

10 

444 
542 
640 

454 
552 
650 

464 
562 
660 

473 
572 
670 

483 

582 
680 

493 
591 
689 

503 
601 
699 

513 
611 
709 

528 

621 
719 

1  532 
631 
729 

10 
10 
10 

444 
445 
446 

788 
836 
933 

748 
846 
943 

758 
856 
953 

768 
865 
963 

777 
875 
972 

787 
885 
982 

797 
895 
992 

807 

904 

*002 

816 

914 

*011 

826 

924 

*021 

10 
10 
10 

447 
448 
449 

450 

451 
452 
453 

65  031 
128 
225 

040 
137 
234 

050 
147 
244 

060 
157 
254 

070 
167 
263 

079 
176 
273 

089 
186 
283 

099 
196 
292 

108  j  118 
205  215 
302  1  312 

10 
10 
10 

321 

331 

341 

350 

360 

369 

379 

389 

398  408 

10 

418 
514 
610 

427 
523 
619 

437 
533 
629 

447 
543 
639 

456 
552 

648 

466 
562 
658 

475 

571 
667 

485 
581 
677 

495  i  504 
591  i  600 
686  1  696 

10 
10 
10 

454 
455 
456 

706 
801 
896 

715 
811 
906 

725 
820 
916 

734 
830 
925 

744 

839 
9:35 

758 

849 
944 

763 
858 
954 

772 
868 
963 

782 
877 
973 

792 

887 
982 

9 
9 
9 

457 
458 
459 

460 

461 
462 
463 

992 

66  087 

181 

*001 
096 
191 

*011 
106 
200 

*020 
115 
210 

*030 
124 

219 

»039 
134 

229. 

*049 
148 
238 

*058 
153 
247 

*068 
162 
257 

*077 
172 
266 

9 

9 
9 

276 

285 

295 

304 

314 

333 

332 

342 

351 

361 

9 

370 
464 
558 

380 

474 
567 

389 
483 
577 

398 
492 
586 

408 
502 
596 

417 
511 
605 

427 
521 
614 

436 
530 
624 

445 
539 
633 

455 
549 
642 

9 
9 
9 

464 
465 
466 

652 
745 
839 

661 
755 

848 

671 
764 

857 

680 
773 
367 

689 
783 

876 

699 
792 

885 

708 
801 
894 

717 
811 
904 

727 
820 
913 

736 
829 
922 

9 
9 
9 

467 
468 
469 

932 

67  025 

117 

941 
034 
127 

950 
043 
136 

960 
052 
145 

969 
062 
154 

978 
071 
164 

987  1 

080 

173 

997 
089 
182 

*006 
099 
191 

*015 
108 
201 

9 
9 
9 

N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

D 

126 


COMMON  LOGARITHMS  OF  NUMBERS. 


N 

O 

1 

2 

3 

4 

5 

6 

7 

8 

9 

D 

470 

471 
472 
473 

210 

219 

228 

237 

247 

256 

265 

274 

284 

293 

9 

302 
394 
486 

311 
403 
495 

321 
413 
504 

330 
422 
514 

339 
431 
523 

348 
440 
532 

357 
449 
541 

367 
459 
550 

376 
468 
560 

385 

477 
569 

9 
9 
9 

474 
475 
476 

578 
669 
761 

587 
679 
770 

596 

688 
779 

605 
697 
788 

614 
706 

797 

624 
715 
806 

633 

724 
815 

642 
733 
825 

651 
742 
834 

660 
752 

843 

9 
9 
9 

477 
478 
479 

480 

481 
483 
483 

852 

943 

68  034 

861 
952 
043 

870 
961 
052 

879 
970 
061 

888 
979 
070 

897 
988 
079 

906 
997 
088 

916 

*006 
097 

925 

*015 

106 

934 

*024 
115 

9 
9 
9 

124 

133 

142 

151 

160 

169 

178 

187 

196 

205 

9 

215 
305 
395 

224 
314 
404 

233 
323 
413 

242 
332 
422 

251 
341 
431 

260  269 
850  i  359 
440  449 

278 
368 
458 

287 
377 
467 

296 
386 
476 

9 
9 
9 

484 
485 
486 

485 
574 
664 

494 
583 
673 

502 
592 
681 

511 
601 
690 

520 
610 
699 

529 
619 

708 

538 
628 

717 

547 
637 
726 

556 
646 
735 

565 
655 

744 

9 
9 
9 

487 
488 
489 

490 

491 
492 
493 

753 

842 
931 

763 
851 
940 

771 
860 
949 

780 
869 
958 

789 
878 
966 

797 
886 
975 

806 
895 
984 

815 
904 
993 

824 

913 

*002 

833 

922 

*011 

9 
9 
9 

69  020 

028 

037 

046 

055 

064 

073 

082 

090 

099 

9 

108 
197 

285 

117 
205 
294 

126 
214 
302 

135 
223 
311 

144 
232 
320 

152 

241 
329 

161 
249 
338 

170 
258 
346 

179 
267 
355 

188 
276 
364 

9 
9 
9 

494 
495 
496 

373 
461 
548 

381 
469 
557 

390 

478 
566 

399 
487 
574 

408 
496 
583 

417 
504 
592 

425 
513 
601 

434 
522 
609 

443 

531 
618 

452 
539 
627 

9 
9 
9 

497 
498 
499 

500 

501 
502 
503 

636 
723 
810 

644 

732 
819 

653 

740 

827 

662 

749 
836 

671 

758 
845 

679 

767 
854 

688 
775 
862 

697 

784 
871 

705 
793 
880 

714 
801 

888 

•  9 
9 
9 

897 

906 

914 

923 

932 

940 

949 

958 

966 

975 

9 

984 

70  070 

157 

992 
079 
165 

*001 
088 
174 

*010 
096 
183 

*018 
105 
191 

*027 
114 
200 

*086 
122 
209 

*044 
131 

217 

*053 
140 
226 

*062 
148 
234 

9 
9 
9 

504 
505 
506 

243 
329 
415 

252 

338 
424 

260 
346 
432 

269 
355 
441 

278 
364 
449 

286 
372 
458 

295 
381 
467 

303 
389 
475 

312 

398 

484 

321 
406 
493 

9 
9 
9 

507 
508 
509 

501 
586 
672 

509 
595 
680 

518 
603 
689 

526 

612 
697 

535 

621 
706 

544 
629 

714 

552 
638 
723 

561 
646 
731 

569 
655 
740 

578 
663 
749 

9 
9 
9 

N 

O 

1 

2 

3 

4 

5 

6 

7 

8 

9 

D 

COMMON  LOGARITHMS  OF  NUMBERS. 


127 


N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

D 

510 

511 
512 
513 

757 

766 

774 

783 

791 

800 

808 

817 

825 

834 

9 

842 

927 

71012 

851 
935 
020 

859 
944 
029 

868 
952 
037 

876 
961 
046 

885 
969 
054 

893 
978 
063 

902 
986 
071 

910 
995 
079 

919 

*003 

088 

9 
9 

8 

514 
515 
516 

096 
181 
265 

105 
189 
273 

113 

198 
282 

122 

206 
290 

130 
214 

299 

139 
223 
307 

147 
231 
315 

155 
240 
324 

164 

248 
332 

172 
257 
341 

8 
8 
8 

517 
518 
519 

520 

521 
52i 
523 

349 
433 

517 

357 
441 
525 

366 

4§0 
533 

374 
458 
542 

383 
466 
550 

391 
475 
559 

399 
483 
567 

408 
492 
575 

416 

500 

584 

425 

508 
592 

8 
8 
8 

600 

609 

617 

625 

634 

642 

650 

659 

667   675 

8 

684 
767 
850 

692 

775 
858 

700 

784 
867 

709 

792 

875 

717 

800 
883 

725 
809 
892 

734 
817 
900 

742 
825 
908 

750 
834 
917 

759 
842 
925 

8 
8 
8 

524 
525 
526 

933 

72  016 

099 

941 
024 
107 

950 
032 
115 

958 
041 
123 

966 
049 
132 

975 
057 
140 

983 
066 
148 

991 
074 
156 

999 
082 
165 

^008 
090 
173 

8 
8 
8 

527 
528 
529 

530 

531 
532 
533 

181 
263 
346 

189 
272 
354 

198 
280 
362 

206 
288 
370 

214 
296 
378 

222 
304 

387. 

230 
313 
395 

239 
321 
403 

247 

329 
411 

255 

337 
419 

8 
8 
8 

428 

486 

444 

452 

460 

469 

477 

485 

493 

501 

8 

509 
591 
673 

518 
599 
681 

526 
607 
689 

534 
616 
697 

542 
624 

705 

550 
632 
713 

558 
640 

722 

567 
648 
730 

575 
656 
738 

583 
665 
746 

8 
8 
8 

534 
535 

536 

754 
835 
916 

763 
843 
925 

770 
852 
933 

779 
860 
941 

787 
868 
949 

795 
876 
957 

803 
884 
965 

811 
892 
973 

819 
900 
981 

827 
908 
989 

8 
8 
8 

537 
538 
539 

540 

541 
542 
543 

997 

73  078 

159 

*006 
086 
167 

*014 
094 
175 

*022 
102 
183 

*030 
111 
191 

*038 
119 
199 

*046 

127 
207 

*054 
135 
215 

*062 
143 
223 

*070 
151 

231 

8 
8 
8 

239 

247 

255 

263 

272 

280 

288 

296 

804 

312 

8 

320 
400 
480 

328 
408 

488 

336 
416 
496 

344 
424 
504 

352 
432 
512 

360 
440 
520 

368 

448 
528 

376 
456 
536 

384 
464 
544 

392 
472 
552 

8 
8 
8 

544 
545 
546 

560 
640 
719 

568 

648 

727 

576 
656 

735 

584 
664 
743 

592 
672 
751 

600 
679 
759 

608 
687 
767 

616 
695 

775 

624 
703 

783 

632 
711 
791 

8 
8 
8 

547 
548 
549 

799 
878 
957 

807 
886 
965 

815 
894 
973 

823 
902 
981 

830 
910 
989 

838 
918 
997 

846 

926 

*005 

854 

933 

*013 

862 

941 

*020 

870 

949 

*028 

8 
8 
8 

N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

D 

128 


COMMON  LOGARITHMS  OF  NUMBERS. 


N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

D 

550 

551 
552 
553 

74  036 

044 

052 

060 

068 

076 

084 

092 

099 

107 

8 

115 
194 
273 

123 
202 

280 

131 
210 

288 

139 
218 
296 

147 
225 
304 

155 
233 

312 

162 
241 
320 

170 
249 

327 

178 
257 
335 

186 
265 
343 

8 
8 
8 

554 
555 

556 

351 
429 

507 

359 
437 
515 

367 
445 
523 

374 
453 
531 

382 
461 
539 

390 
468 
547 

398 
476 
554 

406 

484 
562 

414 
492 
570 

421 

500 

578 

8 

8 
8 

557 
558 
559 

560 

561 
56-3 
563 

586 
663 
741 

593 
671 

749 

601 
679 

757 

609 
687 
764 

617 
695 

772 

624 

702 

780 

632 
710 

788 

640 
718 
796 

648 
726 
803 

656 
733 
811 

8 
8 
8 

819 

827 

904 
981 
059 

834 

842 

850 

858 

865 

873 

881 

889 

8 

896 

974 

75  051 

912 
989 
066 

920 
997 
074 

927 

*005 

082 

935 

3^012 

089 

943 

*020 

097 

950 

*028 

105 

958 

*035 

113 

966 

*043 

120 

8 
8 
8 

564 
565 
566 

128 
205 

282 

136 
213 

289 

143 
220 
297 

151 

228 
305 

159 
236 
312 

166 
243 
320 

174 
251 
328 

182 
259 
335 

189 
266 
343 

197 
274 
351 

8 
8 
8 

567 
568 
569 

570 

571 
572 
573 

358 

435 
511 

366 
442 
519 

374 

450 
526 

381 

458 
534 

389 
465 
542 

397 
473 
549 

404 
481 
557 

412 

488 
565 

420 

496 

572 

427 
504 
580 

8 
8 
8 

587 

664 
740 

815 

595 

671 

747 
823 

603 

610 

618 

626 

633 

641 

648 

656 

8 

679 
755 

831 

686 
762 

838 

694 

770 
846 

702 

778 
853 

709 

7a5 

861 

717 
793 
868 

724 
800 

876 

732 
808 
884 

8 
8 
8 

574 
575 
576 

891 

967 

76  042 

899 
974 
050 

906 
982 
057 

914 

989 
065 

921 
997 
072 

929 

*005 

080 

937 

*012 
087 

944 

*020 
095 

952 

*027 

103 

959 

*035 

110 

8 
8 
8 

577 
578 
579 

580 

581 
582 
583 

118 
193 

268 

125 
200 

275 

133 

208 
283 

140 
215 

290 

148 
223 
298 

155 

230 
305 

163 
238 
313 

170 
245 
320 

178 
253 
328 

185 
260 
335 

8 
8 
8 

343 

350 

358 

365 

373 

380 

388 

395 

403 

410 

8 

418 
492 
567 

425 
500 
574 

433 
507 

582 

440 
515 

589 

448 
522 

597 

455 
530 
604 

462 
537 

612 

470 
545 
619 

477 
552 
626 

485 
559 
634 

7 
7 
7 

584 

585 
586 

641 
716 
790 

649 
723 

797 

656 

730 

805 

664 

738 
812 

671 
745 
819 

678 
753 

827 

686 
760 
834 

693 
768 
843 

701 
775 
849 

708 
782 
856 

7 
7 
7 

587 
588 
589 

864 

938 

77  012 

871 
945 
019 

879 
953 
026 

886 
960 
034 

893 
967 
041 

901 
975 
048 

908 
982 
056 

916 
989 
063 

923 
997 
070 

930 

*004 

078 

7 
7 
7 

N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

D 

COMMON  LOGARITHMS  OF  NUMBERS. 


129 


N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

D 

590 

591 
592 
593 

085 

093 

100 

107 

115 

122 

129 

137 

144 

151 

7 

159 
232 
305 

166 
240 
313 

173 
247 
320 

181 
254 

327 

188 
262 
335 

195 
269 
342 

203 
276 
349 

210 
283 

357 

217 
291 
364 

225 

298 
371 

7 
7 
7 

594 
595 
596 

379 
452 
525 

386 
459 

532 

393 
466 
539 

401 
474 
546 

408 
481 
554 

415 
488 
561 

422 
495 
568 

430 
503 
576 

437 
510 
583 

444 
517 
590 

7 
7 
7 

597 
598 
599 

600 

601 
602 
603 

597 
670 
743 

605 
677 
750 

612 

685 

757 

619 
692 
764 

627 
699 

772 

634 

706 
779 

641 
714 

786 

648 
721 
793 

656 

728 
801 

663 
735 

808 

7 
7 
7 

815 

822 

mo 

837 

844 

851 

859 

866 

873 

880 

7 

887 

960 

78  032 

895 
967 
039 

902 
974 
046 

909 
981 
053 

916 
988 
061 

924 
996 

068 

931 

*003 

075 

938 

*010 

082 

945 

*017 

089 

952 

*025 

097 

7 
7 
7 

604 
605 
606 

104 
176 
247 

111 
183 
254 

118 
190 
262 

125 
197 
269 

132 

204 
276 

140 
211 
283 

147 
219 
290 

154 
226 

297 

161 
233 
305 

168 
240 
312 

7 
7 
7 

607 
608 
609 

610 

611 
612 
613 

319 
390 
462 

326 
398 
469 

333 
405 
476 

340 
412 
483 

347 
419 
490 

355 

426 
497 

362 
433 
504 

369 
440 
512 

376 
447 
519 

383 
455 
526 

7 
7 
7 

533 

540 

547 

554 

561 

569 

576 

583 

590 

597 

7 

604 
675 
746 

611 
682 
753 

618 
689 
760 

625 
696 

767 

633 

704 

774 

640 
711 
781 

647 

718 
789 

654 

725 
796 

661 
732 
803 

668 
739 
810 

7 
7 
7 

614 
615 
616 

817 
888 
958 

824 
895 
965 

831 
902 
972 

838 
909 
979 

845 
916 
986 

852 
923 
993 

859 

930 

*000 

866 

937 
*007 

873 
944 

*014 

880 

951 

*021 

7 
7 
7 

617 
618 
619 

620 

621 
622 
623 

79  029 
099 
169 

036 
106 
176 

043 
113 
183 

050 
120 
190 

057 
127 
197 

064 
134 

204 

071 
141 
211 

078 
148 
218 

085 
155 

225 

092 
162 
232 

7 
7 
7 

239 

246 

253 

260 

267 

274 

281 

288 

295 

302 

7 

309 
379 
449 

316 
386 
456 

323 
393 
463 

330 
400 
470 

337 
407 

477 

344 
414 
484 

351 
421 
491 

358 
428 
498 

365 
435 

505 

372 
442 

511 

7 
7 
7 

624 
625 
626 

518 
588 
657 

525 

595 
664 

532 
602 
671 

539 
609 
678 

546 
616 

685 

553 
623 
692 

560 
630 
699 

567 
637 

706 

574 
644 

713 

581 
650 
720 

7 
7 

7 

627 
628 
629 

727 
796 
865 

734 

803 

872 

741 
810 
879 

748 
817 
886 

754 
824 

893 

761 
831 
900 

76S 
837 
906 

775 
844 
913 

782 
851 
920 

789 

858 
927 

7 
7 
7 

N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

D 

130 


COMMON   LOGARITHMS  OF  NUMBERS. 


N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

D 

630 

631 
633 
633 

934 

941 

948 

955 

962 

969 

975 

983 

989 

996 

7 

80  003 
073 
140 

010 
079 
147 

017 
085 
154 

034 
093 
161 

080 
099 
168 

087 
106 
175 

044 
113 

182 

051 
120 
188 

058 
127 
195 

065 
134 

202 

7 
7 

634 
635 
636 

209 
377 
346 

216 

384 
353 

233 
391 
359 

339 

398 
366 

236 
305 
373 

243 

312 
380 

250 

318 
387 

257 

335 
393 

264 

332 
400 

271 
339 
407 

7 
7 

637 
638 
639 

640 

641 
642 
643 

414 

483 
550 

421 
489 
557 

438 
496 
564 

434 
502 
570 

441 
509 

577 

448 
516 

584 

455 
533 
591 

463 
530 

598 

468 
536 
604 

475 
543 
611 

7 
7 
7 

618 

635 

633 

638 

645 

653 

659 

665 

673 

679 

747 
814 
883 

7 

7 
7 
7 

686 
754 
831 

693 
760 

838 

699 
767 
835 

706 

774 
841 

713 

781 
848 

730 
787 
855 

736 
794 

862 

733 

801 

868 

740 
808 

875 

644 
645 
646 

889 

956 

81033 

895 
963 
030 

902 
969 
037 

909 
976 
043 

916 
983 
050 

932 
990 
057 

929 
996 
064 

986 

*003 
070 

943 

*010 

077 

949 

*017 

084 

7 
7 
7 

647 
648 
649 

650 

651 
653 
653 

090 
158 
334 

097 
164 
231 

104 
171 

238 

111 
178 
245 

117 

184 
251 

134 

191 

258 

131 

198 
265 

137 
204 
271 

144 
311 

378 

151 

318 

285 

7 
7 
7 

291 

298 

305 

311 

318 

325 

331 

338 

845 

351 

7 

358 
435 

491 

365 
431 

498 

371 

438 
505 

378 
445 

511 

385 
451 
518 

391 
458 
525 

398 
465 
531 

405 
471 
538 

411 
478 
544 

418 
485 
551 

7 
7 
7 

654 
655 
656 

558 
690 

564 
631 
697 

571 
637 
704 

578 
644 
710 

584 
651 
717 

591 
657 
723 

598 
664 
730 

604 
671 

737 

611 
677 
743 

617 

7 
7 
7 

657 
658 
659 

660 

661 
663 
663 

757 
833 
889 

763 
839 
895 

770 

836 
902 

776 

843 
908 

783 
849 
915 

790 
856 
931 

796 

862 
928 

803 
869 
935 

809 

875 
941 

816 
883 
948 

'7 

7 
7 

954 

961 

968 

974 

981 

9S7 

994 

*000 

*007 

*014 

7 

83  030 
086 
151 

027 
093 
158 

033 
099 
164 

040 
105 
171 

046 
113 

178 

053 
119 
184 

060 
125 
191 

066 
132 
197 

073 
138 
204 

079 
145 
210 

7 
7 
7 

664 
665 
666 

317 
382 
347 

823 
389 
854 

230 
295 
360 

236 

302 
367 

243 
308 
378 

249 
315 
880 

256 
331 

387 

263 

328 
393 

269 
334 
400 

276 
341 
406 

7 
7 
7 

667 
668 
669 

413 
478 
543 

419 
484 
549 

436 
491 
556 

432 
497 
562 

439 
504 
569 

445 
510 
575 

453 
517 

583 

458 
523 

588 

465 
530 
595 

471 
536 
601 

7 
7 
7 

N 

0 

1 

2 

3 

4 

5 

6 

7 

« 

9 

D 

COMMON  LOGARITHMS  OF  NUMBERS. 


131 


N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

D 

670 

607 

614 

620 

627 

633 

640 

646 

653 

659 

666 

7 

671 
«73 
673 

672 
737 
802 

679 
743 

808 

685 
750 
814 

693 
756 

831 

698 
763 
837 

705 
769 

834 

711 
776 
840 

718 
783 
847 

734 
789 
853 

730 
795 
860 

6 

6 
6 

674 
675 
676 

866 
930 
995 

872 

937 

*001 

879 

943 

*008 

885 

950 

*014 

892 

956 

*020 

898 

963 

*027 

905 

969 

*033 

911 

975 

*040 

918 

982 

*^046 

924 

988 
*052 

6 

6 
6 

677 
678 
679 

680 

681 
683 
683 

83  059 
133 

187 

065 
129 
193 

072 
136 
200 

078 
142 
206 

085 
149 
213 

091 
155 
219 

097 
161 
235 

104 
168 

233 

110 
174 

238 

117 
181 
345 

6 
6 
6 

251 

257 

264 

270 

276 

383 

289 

296 

302 

308 

6 

315 
378 
442 

331 

385 
448 

327 
391 
455 

834 

398 
461 

340 
404 
467 

347 
410 
474 

353 

417 
480 

359 
433 

487 

366 
429 
493 

372 
436 
499 

6 
6 
6 

684 
6&5 
686 

506 
569 
632 

512 
575 
639 

518 

582 
645 

535 

588 
651 

531 
594 
658 

537 
601 
664 

544 
607 
670 

550 
613 
677 

556 
620 
683 

563 
626 
689 

6 
6 
6 

687 
688 
689 

690 

691 
693 
693 

696 
759 

832 

702 
765 
828 

708 
771 
835 

715 

778 
841 

731 

784 
847 

727 
790 
853 

734 

797 
860 

740 
803 
866 

746 

809 

873 

753 

816 
879 

6 
6 
6 

885 

891 

897 

904 

910 

916 

923 

929 

935 

943 

6 

948 

84  011 

073 

954 

017 
080 

960 
023 
086 

967 
039 
092 

973 
036 
098 

979 
042 
105 

985 
.048 
111 

992 
055 
117 

998 
061 
123 

*004 
067 
130 

6 

6 
6 

604 
695 
690 

136 
198 
261 

142 

205 
267 

148 
211 
273 

155 
217 
280 

161 
223 

286 

167 
230 
292 

173 

236 
298 

180 
242 
305 

186 
248 
311 

192 
255 

317 

6 
6 
6 

697 
698 
699 

700 

701 
702 
703 

323 

386 
448 

330 
392 
454 

336 
398 
460 

342 
404 
466 

348 
410 
473 

354 

417 
479 

361 
423 

485 

367 
439 
491 

373 

435 
497 

379 
442 
504 

6 
6 
6 

510 

516 

578^ 
640 

702 

532 

528 

535 

541 

547 

553 

559 

566 

6 

572 
634 
696 

584 
646 

708 

590 
652 

714 

597 

658 
720 

603 
665 

726 

609  , 
671  ' 
733  j 

615 

677 
739 

621 
683 
745 

638 
689 
751 

6 
6 
6 

704 
705 
706 

757 
819 
880 

763 

825 
887 

770 
831 
893 

776 
837 
899 

782 
844 
905 

788 
850 
911 

794 
856 
917 

800 
862 
934 

807 
868 
930 

813 
874 
936 

6 
6 
6 

707 
708 
709 

942 

85  003 

065 

948 
009 
071 

954 
016 
077 

960 
022 
083 

967 
028 
089 

973 
034 
095 

979 
040 

101  1 

1 

985 
046 
107 

991 
052 
114 

997 
058 
120 

6 
6 
6 

N 

0 

1 

2  1  3 

4 

5 

6 

7 

8 

9 

D 

132 


COMMON  LOGARITHMS  OF  NUMBERS. 


N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

D 

710 

711 
712 
713 

136 

132 

138 

144 

150 

156 

163 

169 

175 

181 

6 

187 
248 
309 

193 
254 
315 

199 
260 
321 

205 
266 
327 

211 

273 
333 

217 
278 
339 

234 

285 
345 

230 
291 
352 

236 

297 
358 

242 
303 
364 

6 
6 
6 

714 
715 
716 

370 
431 
491 

376 

437 
497 

383 
443 
503 

388 
449 
509 

394 
455 
516 

400 
461 
523 

406 
467 
528 

412 
473 
534 

418 
479 
540 

425 

485 
546 

6 
6 
6 

717 
718 
719 

720 

721 
723 
723 

552 

612 
673 

558 
618 
679 

564 

625 

685 

570 
631 
691 

576 
637 

697 

582 
643 
703 

588 
649 
709 

594 
655 

715 

600 
661 
721 

606 
667 

737 

6 
6 
6 

733 

739 

745 

751 

757 

763 

769 

775 

781 

788 

6 

794 
854 
914 

800 
860 
920 

806 
866 
926 

813 
872 
932 

818 
878 
938 

834 
884 
944 

830 
890 
950 

836 
896 
956 

842 
902 
962 

848 
908 
968 

6 
6 
6 

734 
725 
726 

974 

86  034 

094 

980 
040 
100 

986 
046 
106 

992 
053 
112 

998 
058 
118 

^004 
064 
124 

*010 
070 
130 

*016 
076 
136 

*022 
083 
141 

*028 
088 
147 

6 
6 
6 

737 
738 
729 

730 

731 
733 
733 

153 
213 
273 

159 
219 
279 

165 
225 

285 

171 
231 
291 

177 
337 
297 

183 
243 
303 

189 
249 
308 

195 
355 

314 

201 
261 
320 

207 
367 
326 

6 
6 
6 

332 

338 

344 

350 

356 

362 

368 

374 

380 

386 

6 

392 
451 

510 

398 
457 
516 

404 
463 
532 

410 
469 
528 

415 
475 
534 

421 
481 
540 

427 

487 
546 

433 
493 
552 

439 

499 

558 

445 
504 

564 

6 
6 
6 

734 
735 
736 

570 
629 
688 

576 
635 
694 

581 
641 
700 

587 
646 
705 

593 

653 
711 

599 
658 
717 

605 
664 
723 

611 
670 

729 

617 
676 

735 

623 

682 
741 

6 
6 
6 

737 

738 
739 

740 

741 
742 
743 

747 
80(i 
864 

753 

813 
870 

759 

817 
876 

764 

833 

882 

770 
829 

888 

776 
835 
894 

782 
841 
900 

788 
847 
906 

794 
853 
911 

800 
859 
917 

6 
6 
6 

6 

923 

929 

935 

941 

947 

953 

958 

964 

970 

976 

983 

87  040 

099 

988 
046 
105 

994 
052 
111 

999 
058 
116 

*005 
064 
122 

*011 
070 
128 

*017 
075 
134 

*023 
081 
140 

*029 
087 
146 

*035 
093 
151 

6 
6 
6 

744 
745 
746 

157 

216 
274 

163 
331 

280 

169 

237 
286 

175 
233 

291 

181 
339 
297 

186 
245 
303 

192 
251 
309 

198 
256 
315 

204 
262 

330 

210 
268 
326 

6 
6 
6 

747 
748 
749 

332 
390 
448 

338 
396 
454 

344 
402 
460 

349 
408 
466 

355 
413 
471 

361 
419 

477 

367 
435 
483 

373 

431 
489 

379 
437 
495 

384 
442 
500 

6 
6 
6 

N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

D 

COMMON  LOGARITHMS  OF  NUMBERS. 


133 


N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

D 

750 

751 
753 
753 

506 

512 

518 

523 

529 

535 

541 

547 

552 

558 

6 

564 
622 
679 

570 
628 
685 

576 

638 
691 

581 
639 
697 

587 
645 
703 

593 
651 

708 

599 
656 
714 

604 
662 
720 

610 
668 
726 

616 
674 
731 

6 
6 
6 

754 
755 
756 

737 
795 
853 

743 

800 

858 

749 
806 
864 

754 
812 
869 

760 

818 
875 

766 
823 
881 

772 
829 
887 

777 
835 
892 

783 
841 
898 

789 
846 
904 

6 
6 
6 

757 
758 
759 

760 

761 
762 
763 

910 

967 

88  024 

915 
973 
030 

921 

978 
036 

927 
984 
041 

933 
990 
047 

938 
996 
053 

944 

*001 
058 

950 

*007 

064 

955 

*013 

070 

961 

*018 
076 

6 
6 
6 

081 

087 

093 

098 

104 

110 

116 

121 

127 

183 

6 

138 
195 
252 

144 

201 
258 

150 
207 
264 

156 
213 

270 

161 

218 
275 

167 
224 
281 

173 
230 

287 

178 
235 
292 

184 
241 
298 

190 
247 
304 

6 
6 
6 

764 
765 
766 

309 
366 
423 

315 

372 
429 

321 
377 
434 

326 
383 
440 

332 
389 
446 

338 
395 
451 

843 
400 
457 

349 
406 
463 

355 
412 

468 

360 
417 

474 

6 
6 
6 

767 
768 
769 

770 

771 
772 
773 

480 
536 
593 

485 
542 
598 

491 
547 

604 

497 
553 
610 

502 
559 
615 

508 
564 
621 

513 
570 

627 

519 
576 
632 

525 
581 
638 

530 

587 
643 

6 
6 
6 

649 

655 

660 

666 

672 

677 

683 

689 

694 

700 

6 

705 
762 
818 

711 

767 
824 

717 
773 

829 

722 
779 
835 

728 
784 
840 

734 
790 
846 

739 
795 

852 

745 

801 
857 

750 

807 
863 

750 
812 
868 

6 
6 
6 

774 
775 
776 

874 
930 
986 

880 
936 
992 

885 
941 
997 

891 

947 

*003 

897 

953 

*009 

902 

958 

*014 

908 

9(i4 

*020 

913 

969 

*025 

919 

975 

*031 

925 

981 
*037 

6 
6 
6 

777 
778 
779 

780 

781 
782 
783 

89  042 
098 
154 

048 
104 
159 

053 
109 
165 

059 
115 
170 

064 
120 
176 

070 
136 
183 

076 
131 

187 

081 
137 
193 

087 
143 
198 

092 
148 
204 

6 
6 
6 

309 

215 

231 

236 

232 

337 

343 

248 

254 

260 

6 

265 
321 
376 

271 
326 
382 

276 
332 

387 

282 
337 
393 

287 
343 
398 

293 
848 
404 

298 
354 
409 

304 
360 
415 

310 
365 
421 

315 
371 
426 

6 
6 
6 

784 
785 
786 

432 
487 
542 

437 
492 

548 

448 
498 
553 

448 
504 
559 

454 

509 
564 

459 
515 
570 

465 
520 
575 

470 
526 
581 

476 
531 
586 

481 
537 
592 

6 
6 
6 

787 
788 
789 

597 
653 
708 

603 
658 
713 

609 
664 
719 

614 
669 
724 

620 
675 
730 

625 
680 
735 

631 
686 
741 

636 
691 
746 

642 
697 
752 

647 

702 

757 

6 
6 
6 

N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

D 

134 


COMMON  LOGARITHMS  OF  NUMBERS. 


N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

D 

790 

791 
792 
793 

763 

768 

774 

779 

785 

790 

796 

801 

807 

812 

5 

818 
873 
927 

823 
878 
933 

829 

883 
938 

834 
889 
944 

840 
894 
949 

845 
900 
955 

851  !  856 
905  911 
960  966 

862 
916 
971 

867 
922 
977 

5 
5 
5 

794 
795 

796 

982 

90  037 

091 

988 
042 
097 

993 
048 
102 

998 
053 
108 

*004 
059 
113 

»009 
064 
119 

*015 
069 
124 

*020 
075 
129 

*026 
080 
135 

*031 

086 
140 

5 
5 
5 

797 
798 
799 

800 

801 
802 
803 

146 
200 
255 

151 

206 
260 

157 
21] 
266 

162 
217 
271 

168 
222 

276 

173 

227 
282 

179 
233 

287 

184 
238 
293 

189 
244 
298 

195 

249 
304 

5 
5 
5 

809  1  314 

320 

325 

331 

336 

342 

347 

352 

358 

5 

363 

417 
472 

369 
423 

477 

374 

428 
482 

380 
434 

488 

385 
439 
493 

390 
445 

499 

396 
450 
504 

401 
455 
509 

407 
461 
515 

412 
466 
520 

5 
5 
5 

804 
805 
806 

526 
580 
634 

531 

585 
639 

536 
590 
644 

542 
596 
650 

547 
601 
655 

553 

607 
660 

558 
612 
666 

563 

617 
671 

569 
623 
677 

574 

628 
682 

5 

5 
5 

807 
808 
8U9 

810 

811 
812 
813 

687 
741 
795 

693 

747 
800 

698 

752 
806 

703 
757 
811 

709 
763 
816 

714 
768 
822 

720 
773 

827 

725 

779 

832 

730 

784 
838 

736 
789 
843 

5 
5 
5 

849 

854 

859 

865 

870 

875 

881 

886 

891 

897 

5 

902 

956 

91009 

907 
961 
014 

913 
966 
020 

918 
972 
025 

924 

977 
030 

929 
982 
036 

934 
988 
041 

940 
993 
046 

945 

998 
052 

950 

*004 

057 

5 
5 

5 

814 
815 
816 

062 
116 
169 

068 
121 
174 

073 
126 
180 

078 
132 

185 

084 
137 
190 

089 
142 
196 

094 
148 
201 

100 
153 
206 

105 

158 
212 

110 
164 
217 

5 
5 
5 

817 
818 
819 

820 

821 
822 
823 

222 

275 
328 

228 
281 
334 

233 
2S6 
339 

238 
291 
344 

243 
297 
350 

249 
302 
355 

254 
307 
360 

259 
312 
365 

265 

318 
371 

270 
323 
376 

•5 
5 
5 

381 

387 

392 

397 

403 

408 

413 

418 

424 

429 

5 

434 

487 
540 

440 
492 
545 

445 
498 
551 

450 
503 
556 

455 
508 
561 

461 
514 
566 

466 
519 
572 

471 
524 

577 

477 
529 
582 

482 
535 

587 

5 
5 
5 

824 

825 
826 

593 
645 
698 

598 
651 
703 

603 
656 
709 

609 
661 
714 

614 
666 
719 

619 
672 

724 

624 
677 
730 

630 
682 

735 

635 
687 
740 

640 
693 
745 

5 
5 
5 

827 
828 
829 

751 
803 
855 

756 
808 
861 

761 
814 
866 

766 
819 
871 

772 
824 
876 

777 
829 
882 

782 
834 

887 

787 
840 
892 

793 
845  1 
897 

798 
850 
903 

5 
5 
5 

N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

D 

COMMON  LOGARITHMS  OF  NUMBERS. 


136 


N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

D 

830 

831 
832 
833 

908 

913 

918 

924 

929 

934 

939 

944 

950 

955 

5 

960 

92  012 

065 

965 
018 
070 

971 
023 
075 

976 
028 
080 

981 
033 

085 

986 
038 
091 

991 
044 
096 

997 
049 
101 

*002 
054 
106 

*007 
059 
111 

5 
5 

5 

834 
835 
836 

117 
169 
221 

122 
174 
226 

127 
479 
231 

132 
184 
236 

137 
189 
241 

143 
195 
247 

148 
200 
252 

153 

205 
257 

158 
210 
262 

163 

215 
267 

5 
5 
5 

8:37 
838 
839 

840 

841 

843 
843 

273 

324 
376 

278 
330 
381 

283 
335 

387 

288 
340 
392 

293 
345 
397 

298 
350 
402 

304 
355 
407 

309 
361 
412 

314 
366 

418 

319 
371 
423 

5 
5 
5 

438 

433 

438 

443 

449 

454 

459 

464 

469 

474 

526 
578 
629 

5 

5 
5 
5 

480 
531 
583 

485 
536 
588 

490 
542 
593 

495 
547 
598 

500 
552 
603 

505 
557 
609 

511 
562 
614 

516 
567 
619 

521 

572 
624 

844 
845 
846 

634 
686 

737 

639 
691 

742 

645 

696 

747 

650 
701 
752 

655 
706 

758 

660 
711 
763 

665 
716 

768 

670 

722 
773 

675 

727 
778 

681 

732 
783 

5 
5 
5 

847 
848 
849 

850 

851 
852 
853 

788 
840 
891 

793 

845 
896 

901 

804 
855 
906 

809 
860 
911 

814 
865 
916 

819 
870 
921 

824 
875 
927 

829 
881 
932 

834 
886 
937 

5 
5 
5 

942 

947 

952 

957 

962 

967 

973 

978 

983 

988 

5 

993 

93  044 

095 

998 
049 
100 

*003 
054 
105 

*008 
059 
110 

*013 
064 
115 

*018 
069 
120 

*024 
075 
125 

*029 
080 
131 

*034 
085 
136 

*039 
000 
141 

5 
5 
5 

854 
855 
856 

146 
197 
247 

151 

202 
252 

156 

207 
258 

161 
212 
263 

166 
217 
268 

171 
222 
273 

176 

227 
278 

181 
232 

283 

186 
237 

288 

192 
242 
293 

5 

5 
5 

857 
858 
859 

860 

861 

862 
863 

298 
349 
399 

303 
354 
404 

308 
359 
409 

313 
364 
414 

318 
369 
420 

323 
374 
425 

328 
379 
430 

334 

384 
435 

339 
389 
440 

344 
394 
445 

5 
5 
5 

450 

455 

460 

465 

470 

475 

480 

485 

490 

495 

5 

500 
551 
601 

505 
556 
606 

510 
561 
611 

515 
566 
616 

520 
571 
621 

526 
576 
626 

531 

581 
631 

536 
586 
636 

541 
591 
641 

546 
596 
646 

5 
5 
5 

864 
865 
866 

651 
702 

752 

656 

707 

757 

661 
712 
762 

666 
717 
767 

671 

722 
772 

676 

727 
777 

682 
732 

782 

687 
737 
787 

692 

742 
792 

697 

747 
797 

5 
5 
5 

867 
868 
869 

802 
852 
902 

807 
857 
907 

812 
862 
912 

817 
867 
917 

822 
872 
922 

827 
877 
927 

832 

882 
932 

837 
887 
937 

842 
892 
942 

847 
897 
947 

5 
5 
5 

N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

D 

136 


COMMON  L00ARITHM8  OF  NUMBERS. 


N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

D 

870 

952 

957 

962 

967 

972 

977 

982 

987 

993 

997 

5 

871 
872 
873 

94  002 
052 
101 

007 
057 
106 

012 
062 
111 

017 
067 
116 

022 
072 
121 

027 
077 
126 

032 
082 
131 

037 
086 
136 

042 
091 
141 

047 
096 
146 

5 
5 
5 

874 
875 
876 

151 
201 
250 

156 
206 
255 

161 
211 
260 

166 
216 
265 

171 
221 
270 

176 
226 
275 

181 
231 

280 

186 
236 

285 

191 
840 
390 

196 

345 
395 

5 
5 
5 

877 
878 
879 

880 

881 
883 
883 

300 
349 
399 

305 
354 

404 

310 
359 

409 

315 
364 
414 

320 
369 
419 

325 
374 
434 

330 
379 
429 

335 

384 
433 

340 

389 
438 

345 
394 
443 

5 
5 
5 

448 

453 

458 

463 

468 

473 

478 

483 

488 

493 

5 

498 
547 
596 

503 
552 
601 

507 
557 
606 

512 
562 
611 

517 
567 
616 

522 
571 
621 

537 
576 
636 

533 
581 
630 

537 
586 
635 

543 
591 
640 

5 
5 
5 

884 
885 
886 

645 
694 
743 

650 
699 

748 

655 
704 
753 

660 
709 

758 

665 
714 
763 

670 
719 
768 

675 

734 
773 

680 
729 

778 

685 
734 

783 

689 

738 
787 

5 
5 
5 

887 
888 
889 

890 

891 
892 
893 

792 
841 
890 

797 
846 
895 

802 
851 
900 

807 
&56 
905 

812 
861 
910 

817 
866 
915 

832 
871 
919 

837 
876 
934 

833 
880 
939 

836 

885 
934 

5 
5 
5 

939 

944 

949 

954 

959 

963 

968 

973 

978 

983 

5 

988 

95  036 

085 

993 
041 
090 

998 
046 
095 

*002 
051 
100 

*007 
056 
105 

*012 
061 
109 

*017 
066 
114 

*033 
071 
119 

*037 
075 
134 

*033 
080 
139 

5 
5 
5 

894 
895 
896 

134 

182 
231 

139 
187 
236 

143 
192 
240 

148 
197 
245 

153 
202 
250 

158 
207 
255 

163 
211 
260 

168 
316 
265 

173 

331 
370 

177 
336 

374 

5 
5 
5 

897 
898 
899 

900 

901 
902 
903 

279 
328 
376 

284 
332 
381 

289 
337 
386 

294 
342 
390 

299 
347 
395 

303 
352 
400 

308 
357 
405 

313 
361 
410 

318 
366 
415 

333 
371 
419 

•  5 
5 
5 

424 

429 

434 

439 

444 

448 

453 

458 

463 

468 

5 

472 
521 
569 

477 
525 
574 

482 
530 
578 

487 
535 
583 

492 
540 

588 

497 
545 
593 

501 
550 
598 

506 
554 
603 

511 
559 

607 

516 
564 
613 

5 
5 
5 

904 
905 
906 

617 
665 
713 

622 
670 

718 

626 
674 

722 

631 

679 

727 

636 
684 
732 

641 

689 
737 

646 
694 

742 

650 
698 
746 

655 

703 
751 

660 
708 
756 

5 
5 
5 

907 
908 
909 

761 
809 
856 

766 
813 
861 

770 
818 
866 

775 
823 
871 

780 

828 
875 

785 
832 
880 

789 
837 
885 

794 

843 
890 

799 
847 
895 

804 
853 
899 

5 
5 
5 

N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

D 

COMMON  LOGARITHMS  OF  NUMBERS. 


137 


N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

D 

910 

911 
913 
913 

904 

909 

914 

918 

923 

928 

933 

938 

942 

947 

5 

952 

999 
96  047 

957 

*004 

052 

961 

*009 

057 

966 

*014 

061 

971 

*019 

066 

976 

*023 

071 

980 

*028 

076 

985 

*033 

080 

990 

*038 
085 

995 

*042 

090 

5 
5 
5 

914 
915 
916 

095 
142 
190 

099 
147 
194 

104 
152 
199 

109 
156 
204 

114 
161 
209 

118 
166 
213 

123 

171 
218 

128 
175 

223 

133 

180 
227 

137 

185 
232 

5 
5 
5 

917 

918 
919 

920 

921 
922 
923 

237 

284 
332 

242 
289 
336 

246 
294 
341 

251 
298 
346 

256 
303 
350 

261 

308 
355 

265 
313 
360 

270 
317 
365 

275 
322 

369 

280 
327 
374 

5 
5 
5 

379 

384 

388 

393 

398 

402 

407 

412 

417 

421 

5 

426 
473 
520 

431 

478 
525 

435 
483 
530 

440 
487 
534 

445 
492 
539 

450 
497 
544 

454 
501 
548 

459 
506 
553 

464 
511 
558 

468 
515 
562 

5 
5 
5 

924 
925 
926 

567 
614 
661 

572 
619 
666 

577 
624 
670 

581 
628 
675 

586 
633 
680 

591 

638 

685 

595 
642 

689 

600 
647 
694 

605 
652 
699 

609 
656 
703 

5 
5 
5 

927 
628 
929 

930 

931 
932 
933 

708 
755 

802 

713 
759 
806 

717 
764 
811 

722 

769 
816 

727 
774 
820 

731 

778 
825 

736 

783 
830 

741 
788 
834 

745 

792 
839 

750 
797 
844 

5 
5 
5 

848 

853 

858 

862 

867 

872 

876 

881 

886 

890 

5 

895 
942 
988 

900 
946 
993 

904 
951 
997 

909 

956 

*002 

914 

960 
*007 

918 

965 

*011 

923 

970 

*016 

928 

974 

*021 

932 

979 
*025 

937 

984 

*030 

5 
5 
5 

934 
935 
936 

97  035 
081 
128 

039 
086 
132 

044 
090 
137 

049 
095 
142 

053 
100 
146 

058 
104 
151 

063 
109 
155 

067 
114 
160 

072 
118 
165 

077 
123 
169 

5 
5 
5 

937 
938 
939 

940 

941 
942 
943 

174 

220 
267 

179 
225 

271 

183 

230 
276 

188 
234 
280 

192 
239 

285 

197 
243 
290 

202 

248 
294 

206 
253 
299 

211 
257 
304 

216 
262 
308 

5 
5 
5 

313 

317 

322 

327 

331 

336 

340 

345 

350 

354 

5 

359 
405 
451 

364 
410 
456 

368 
414 
460 

373 
419 
465 

377 
424 
470 

382 
428 
474 

387 
433 
479 

391 
437 
483 

396 
442 
488 

400 
447 
493 

5 
5 
5 

944 
945 
946 

497 
543 
589 

502 
548 
594 

506 
553 
598 

511 
557 
603 

516 

562 
607 

520 
566 

612 

525 

571 
617 

529 
575 
621 

534 
580 
626 

539 

585 
630 

5 
5 
5 

947 
948 
949 

635 

681 

727 

640 
685 
731 

644 
690 
736 

649 
695 
740 

653 
699 
745 

658 
704 
749 

663 

708 

754 

667 
713 
759 

672 
717 
763 

676 

722 
768 

5 

5 
5 

N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

D 

138 


COMMON  LOGARITHMS  OF  NUMBERS. 


N 

0 

1 

2 

3 

4 

6 

6 

7 

8 

9 

D 

950 

951 
952 
953 

772 

777 

782 

786 

791 

795 

800 

804 

809 

813 

5 

818 
864 
909 

823 

868 
914 

827 
873 
918 

832 
877 
923 

836 

882 
928 

841 

886 
932 

845 
891 
937 

850 
896 
941 

855 
900 
946 

859 
905 
950 

5 
5 
5 

954 
955 
956 

955 

98  000 

046 

959 
005 
050 

964 
009 
056 

968 
014 
059 

973 

019 
064 

978 
023 

068 

982 
028 
073 

987 
032 
078 

991 

037 
082 

996 
041 
087 

5 
5 
5 

957 

958 
959 

960 

961 
962 
963 

091 
137 

182 

096 
141 

186 

100 
146 
191 

105 
150 
195 

109 

155 
200 

114 
159 

204 

118 
164 
209 

123 

168 
214 

127 
173 

218 

132 

177 
223 

5 
5 
5 

227 

232 

236 

241 

245 

250 

254 

259 

263 

268 

^ 

273 
318 
363 

277 
322 
367 

281 
327 
372 

286 
331 
376 

290 
336 
381 

295 
340 
385 

299 
345 
390 

304 
349 
394 

308 
354 
399 

313 

358 
403 

5 
5 
5 

964 
965 
966 

408 
453 
498 

412 
457 
502 

417 
462 
507 

421 
466 
511 

426 

471 
516 

430 
475 
520 

435 
480 
525 

439 

484 
529 

444 

489 
534 

448 
493 
538 

5 
4 
4 

967 
968 
969 

970 

971 
972 
973 

.  543 

588 
632 

547 
592 
637 

552 

597 
641 

556 
601 
646 

561 
605 
650 

565 

610 
655 

570 
614 
659 

574 
619 
664 

579 
623 
668 

583 
628 
673 

4 
4 
4 

677 

682 

686 

691 

695 

700 

704 

709 

713 

717 

4 

722 
767 
811 

726 
771 
816 

731 

776 
820 

735 

780 
825 

740 

784 
829 

744 

789 
834 

749 
793 

838 

753 

798 
843 

758 
802 
847 

762 

807 
851 

4 
4 
4 

974 
975 
976 

856 
900 
945 

860 
905 
949 

865 
909 
954 

869 
914 
958 

874 
918 
963 

878 
923 
967 

883 
927 
972 

887 
932 
976 

892 
936 

981 

896 
941 

985 

4 
4 
4 

977 
978 
979 

980 

981 
982 
983  . 

989 

99  034 

078 

994 
038 
083 

998 
043 

087 

*003 
047 
092 

*007 
052 
096 

*012 
056 
100 

*016 
061 
105 

*021 
065 
109 

*025 
069 
114 

*029 
074 
118 

'  4 
4 
4 

123 

127 

131 

136 

140 

145 

149 

154 

158 

162 

4 

167 
211 
255 

171 
216 

260 

176 
220 
264 

180 
224 
269 

185 
229 
273 

189 
233 

277 

193 

238 
282 

198 
242 
286 

202 
247 
291 

207 
251 
295 

4 
4 
4 

984 
985 
986 

300 
344 

388 

304 
348 
392 

308 
352 
396 

313 
357 
401 

317 
361 
405 

322 
366 
410 

326 
370 

414 

330 
374 
419 

335 
379 
423 

339 
383 

427 

4 
4 

4 

987 
988 
989 

432 
476 
520 

436 
480 
524 

441 
484 
528 

445 

489 
533 

449 
493 

454 
498 
542 

458 
502 
546 

463 
506 
550 

467 
511 
555 

471 
515 
559 

4 
4 
4 

N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

D 

COMMON  LOGARITHMS  OF  NUMBERS. 


139 


N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

D 

990 

991 

564 

568 

5';2 

577 

581 

585 

590 

594 

599 

603 

4 

607 

612 

616 

621 

625 

629 

634 

638 

643 

647 

4 

992 

651 

656 

660 

664 

669 

673 

677 

682 

686 

691 

4 

993 

695 

699 

704 

708 

712 

717 

721 

726 

730 

734 

4 

994 

739 

743 

747 

752 

756 

760 

765 

769 

774 

778 

4 

995 

782 

787 

791 

795 

800 

804 

808 

813 

817 

822 

4 

996 

826 

830 

835 

839 

843 

848 

852 

856 

861 

865 

4 

997 

870 

874 

878 

883 

887 

891 

896 

900 

904 

909 

4 

998 

913 

917 

922 

926 

930 

935 

939 

944 

948 

952 

4 

999 

957 

961 

965 

970 

974 

978 

983 

987 

991 

996 

4 

N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

D 

THE  METRIC  TABLES  OF  WEIGHTS  AND 
MEASURES. 

The  Metric  System  is  a  decimal  system  of  weights  and 
measures. 

The  basis  of  the  whole  system  is  the  metre. 

The  length  of  a  metre  is  defined  by  a  platino-iridium  bar 
kept  in  the  International  Metric  Bureau  at  Paris.  The 
metre  was  meant  to  be  one  ten-millionth  of  the  distance 
from  the  equator  to  the  pole,  but  a  slight  error  in  the 
calculation  has  been  discovered. 

The  Latin  prefixes  indicate  the  denominations  smaller 
than  the  unit,  and  the  Greek  prefixes  the  denominations 
larger  than  the  unit.     Thus  : 


Deci    designates  tenth. 


Centi 

hundredth. 

Milli 

thousandth. 

Deka 

ten. 

Hekto 

hundred. 

Kilo 

thousand. 

Myria 

ten  thousand. 

The  denominations  in  more  frequent  use  are  denoted  by 
heavier  type. 


142  METRIC    TABLES. 

LENGTH. 

TABLE. 

10  millimetres  C^"")  =  1  centimetre  f"'). 

10  centimetres  =  1  decimetre  C""). 

10  decimetres  =  1  metre  (■"). 

10  metres  =  1  dekametre  (^"'). 

10  dekametres  =  1  hektometre  ("™). 

10  hektometres  =  1  kilometre  (^'"). 

10  kilometres  =  1  myriametre  (^'°). 

SURFACE. 

The  units  of  surface  are  squares  whose  dimensions  are 
the  corresponding  linear  units  ;  hence  it  takes  10  times  10, 
or  100,  of  one  denomination  to  make  one  of  the  next 
higher.  For  measuring  small  surfaces  the  principal  unit 
is  the  square  metre. 

TABLE. 

100  square  millimetres  («<!'"'")  =  1  square  centimetre  («<i  <='»). 
100  square  centimetres  =  1  square  decimetre  (^i'^'"). 

100  square  decimetres  =  1  square  metre  (''J'"). 

100  square  metres  =  1  square  dekametre  («'''''°). 

100  square  dekametres  =  1  square  hektometre  (**^"'°). 

100  square  hektometres  =  1  square  kilometre  {^^^'^). 


LAND. 


TABLE. 


100  cen tares  ('^*)  =  1  are  ("). 

100  ares  =  1  hektare  ("*). 

A  centare  is  a  square  metre,  an  are  a  square  deka- 
metre, and  a  hektare  a  square  hektometre. 


METRIC    TABLES,  143 


VOLUME. 

The  units  of  volume  are  cubes  whose  dimensions  are  the 
corresponding  linear  units ;  hence  it  takes  10  times  10 
times  10,  or  1000,  of  one  denomination  to  make  one  of  the 
next  higher. 

TABLE. 

1000  cubic  millimetres  f^"'"'")  =  1  cubic  centimetre  f  "««>). 
1000  cubic  centimetres  =  1  cubic  decimetre  ('^"'^■»). 

1000  cubic  decimetres  =  1  cubic  metre  (*"""). 


WOOD. 

TABLE. 

10  decisteres  (''")  =  1  stere  ('*). 

10  steres  =  1  dekastere  (^*). 

A  stere  is  a  cubic  metre. 


CAPACITY. 

The  unit  of  capacity  is  a  litre,  which  equals  a  cubic  deci- 
metre. 

TABLE. 

10  millilitres  ("')  =  1  centilitre  («'). 
10  centilitres         =  1  decilitre  ('^'). 
10  decilitres  =  1  litre  ('). 

10  litres  =  1  dekalitre  (^'). 

10  dekalitres        =  1  hektolitre  («'). 
10  hektolitres       =  1  kilolitre  (^'). 


144 


METRIC  TABLES. 


WEIGHT. 

The  unit  of  weight  is  a  gram,  which  equals  the  weight 
of  a  cubic  centimetre  of  water  at  its  greatest  density. 


TABLE. 


10  milligrams  f"^) 
10  centigrams 
10  decigrams 
10  grams 
10  dekagrams 
10  hektograms 
1000  kilograms 


=  1  centigram  f^^). 

=  1  decigram  i^^), 

=  1  gram  (s). 

=  1  dekagram  (^^). 

=  1  hektogram  ("^). 

=  1  kilogram,  or  kilo  (^). 

=  1  ton  C)' 


METRIC  EQUIVALENTS. 


1  metre=:39.37  in.  =1.0936  yd. 
1  kilometre  =  .62138  mile 


1  hektare 

1  litre 

1  gram 
1  kilogram 

1  stere 


=  2.471  acres 

_    j  .908  qt.  dry 
~    (  1.0567  qt.  liq. 
=  15.432  grains 
=  2.2046  lbs. 
=  .2759  cord 


1  yard 
1  mile 

1  acre 
1  qt.  dry 
1  qt.  liq. 
1  grain 
1  pound 
1  cord 


=  .9144  m. 
=  1.6093  kilo- 
metres. 
=  .4047  Ha. 
=  1.1011. 
=  .9463  1. 
=  .0648  gram. 
=  .4536  K. 
=  3.625  steres. 


APPROXIMATE  METRIC  EQUIVALENTS. 


1  cm.  =  I  in. 
1  Km.  =  f  mile. 


2|  bush. 
2|  lbs. 
2200  lbs. 


-^IBB^? 


OF  THE 


UNfVERSITY 


OF 


«^UPORN\b< 


LONGMANS,  GREEN,  ^  CO: S  PUBLICATIONS. 


THE  HARPUR  EUCLID.  An  Edition  of  Euclid's 
Elements  revised  in  accordance  with  the  Reports  of  the 
Cambridge  Board  of  Mathematical  Studies  and  the  Ox- 
ford Board  of  the  Faculty  of  Natural  Science. 

By  E.  M.  Langley,  Senior  Mathematical  Master,  the  Modern  School, 
Bedford,  and  W.  S.  Phillips,  M.  A.,  Senior  Mathematical  Master  at 
Bedford'  Grammar  School.  With  numerous  Notes,  and  nearly 
i,ooo  Miscellaneous  Exercises  and  Supplementary  Propositions. 
Complete  edition.  Books  I-VI  and  XI.  (1-21).  Crown  8vo. 
524  pages.     $1.50. 

•^See  Longmans,  Green,  6-  Co: s  Catalogue  of  Educational   Works  for  the 
Separate  Editions  of  the  Various  Books. 

.  .  .  An  attempt  has  been  made  in  the  Notes  and  Miscellaneous  Exercises 
to  familiarise  the  student  with  such  terms  and  ideas  as  he  will  be  likely  to  meet 
with  in  his  higher  reading  and  in  treatises  on  Elementary  Geometry  by  other 
writers,  and  to  indicate  by  difference  of  type  important  theorems  and  problems 
which  should  be  well  known  to  him,  although  not  given  among  Euclid's 
propositions.  At  the  same  time  the  work  is  not  intended  to  be  a  substitute  for 
such  works  as  Casey's  well-known  Sequel  to  Euclid,  but  to  serve  as  an  introduc- 
tion to  them. 

.  .  .  The  short  treatise  "  On  Quadrilaterals  "  will,  it  is  hoped,  be  found  in- 
teresting to  those  students  who  have  mastered  the  previous  exercises,  and  use- 
ful to  teachers  in  supplying  a  large  number  of  easy  and  instructive  exercises  in 
a  short  compass.     .    .     .     [Preface.] 

PRACTICAL  PLANE  AND  SOLID  GEOMETRY. 

By  I.  Hammond  Morris,  South  Kensington  Art  Department.  Fully 
Illustrated  with  Drawings  done  specially  for  the  Book  by  the  Author. 
i2mo.     264  pages.     80  cents. 

The  Volume  treats  of :  i.  Construction  and  Use  of  Plain  Scales  and  Scales 
of  Chords.  2.  Proportional  Division  of  Lines.  3.  Mean  and  4th  Proportional. 
4.  Lines  and  Circles  required  in  drawing  out  Geometrical  Patterns.  5.  Re- 
duction and  Enlargement  of  Plane  Figures.  6.  Polygons  on  Lines  and  in  Cir- 
cles. 7.  Irregular  Polygons.  8.  Irregular  Figures  :  the  Ellipse,  etc.  9.  Plan, 
Elevation,  and  Section  of  Cube,  Pyramid,  Prism,  Cylinder,  Cone,  and  Sphere 
in  Simple  Positions. 

Solid  Geometry.  The  Principles  of  Projection.  Definition  of  Terms,  etc. 
Simple  Problems  relating  to  Lines  and  Planes.  Plan  and  Elevation  of  Simple 
Solids  resting  on  the  Horizontal  Plane,  and  also  when  Inclination  of  Two  Sides 
or  of  Plane  and  One  Side  are  given.  Sections  of  such  Solids  by  Vertical  and 
Horizontal  Planes. 

Graphic  Arithmetic.  The  Representation  of  Numbers  by  Lines.  The  Mul- 
tiplication of  Numbers  by  Construction.  The  Division  of  Numbers  by  Con- 
struction.    The  Determination  of  the  Square  Root  of  Numbers  by  Construction. 


LONGMANS,  GREEN,  &   CO.,  91-93   Fifth  Avenue,   New  York. 


LONGMANS,  GREEN,  ^  CO: S  PUBLICATIONS, 

A  TREATISE  ON  CONIC  SECTIONS.  Contain- 
ing an  Account  of  some  of  the  most  Important 
Modern  Algebraic  and  Geometric  Methods. 

By  G.  Salmon,  D.D.,  F.R.S.     8vo.     416  pages.    $3.75. 

LONGMANS'  ELEMENTARY  TRIGONOMETRY. 

By  Rev.  Frederick  Sparks,  B.  A.,  late  Lecturer  of  Worcester  College, 
Oxford  ;  Mathematical  Tutor,  Manor  House,  Lee.  Crown  8vo. 
192  pages.     80  cents. 

MILWAUKEE   ACADli,...  x  . 

•'  Longmans'  School  Trigonometry  exactly  meets  my  idea  of  what  an  ele- 
mentary Trigonometry  should  be.  ...  I  like  the  prominence  given  to 
purely  mathematical  reasoning  and  the  relegation  of  the  solution  of  triangles 
to  its  proper  place." — J.  HOWARD  Pratt,  Milwaukee,  Wis. 

CASE   SCHOOL   OF    APPLIED    SCIENCE. 

"It  seems  to  be  very  complete,  and  to  abound  in  well-arranged  examples." 
—Professor  Chas.  S.  Howe,  Cleveland.  O. 

"  One  of  the  most  admirable  text-books  we  have  ever  seen.  While  it  i>- 
eminently  fitted  to  its  title,  the  title  is  in  no  way  fitted  to  express  the  high,  r 
merits  of  the  book,  which  is  alike  suited  for  use  in  the  class-room  and  as  a  ref- 
erence book  for  the  engineer.  There  is  a  fulness,  neatness,  precision,  and 
finish  in  all  the  details  that  renders  it  worthy  of  the  highest  praise." — Nation. 

NOTES    ON    TRIGONOMETRY   AND    LOGARITHMS. 

By  the  Rev.  J.  M.  Eustace,  M.A.,  St.  John's  College,  Cambridge. 
i2mo.     311  pages.    $1.35. 

*J^ For  other  books  on  Algebra,    Trigonometry,   etc.,  see  Longmans,  Green,  &> 
Co,'s  Catalogue  of  Educational  Works. 

A   TEXT-BOOK    OF    GEOMETRICAL   DEDUCTIONS. 

By  James  Blaikie,  M.A.,  and  W.  Thomson,  M.A.,  B.Sc.     i2mo. 
Book  I.       Corresponding  to  Euclid,  Book  I.     i2mo.     60  cents. 
Book  n.     Corresponding  to  Euclid,  Book  IL     i2mo.     30  cents. 

ELEMENTARY  GEOMETRY,  CONGRUENT  FIGURES. 

By  O.  Henrici,  Ph.D.,  F.R.S.,  Professor  of  Pure  Mathematics.  Uni- 
versity College,  London.  141  Diagrams.  (London  Science  Class- 
BooKS.)     i6mo.     210  pages.     50  cents. 


LONGMANS,    GREEN,  &  CO.,  91-93  Fifth  Avenue,    New   York. 


LONGMANS,   GREEN,  &-  CO.' S  PUBLICATIONS. 

THE  THEORY  OF  EQUATIONS.     With  an  Intro- 
duction to  the  Theory  of  Binary  Algebraic  Forms. 

By  William  Snow  Burnside,  M.A.,  Fellow  of  Trinity  College,  Dublin  ; 
Erasmus  Smith's  Professor  of  Mathematics  in  the  University  of  Dub- 
lin ;  and  Arthur  William  Panton,  M.A.,  Fellow  and  Tutor  of 
Trinity  College,  Dublin ;  Donegal  Lecturer  in  Mathematics.  (Dub- 
lin University  Press  Series.)    463  pages.     8vo.     $4.50. 

ELEMENTARY  ALGEBRA.     With  Numerous  Ex- 
amples and  Exercises. 

By  Robert  Graham,  M.A.,  T.C.D.,  Ex-Scholar,  Senior  Moderator,  and 
Gold  Medallist  in  Mathematical  Physics.      i2mo.     320  pages.     $1.50. 

The  aim  of  the  writer  has  been  to  give  the  work  necessary  in  an  elementary 
algebra  as  briefly  and  effectively  as  possible,  and  to  devote  more  attention  to 
the  subject  of  B'actors  than  is  usually  given. 

LONGMANS'  ELEMENTARY  ALGEBRA. 

By  W.  S.  Beard,  B.A.,  Mathematical  School,  Rochester.  Crown  8vo. 
163  pages.     50  cents.     With  Answers.     219  pages.     60  cents. 

MILWAUKEE  ACADEMY. 

"  For  elementary  work  in  algebra  I  have  seen  nothing  equal  to  Longmans' 
Elementary  Algebra, " — J.  Howard  Pratt,  Milwaukee,  Wis. 

CAMBRIDGE   ENGLISH   HIGH   SCHOOL. 

"This  algebra  impresses  me  as  a  very  skilfully  prepared  work— rich  in  its 
methods  and  explanations,  in  its  devices  to  awaken  thought  in  the  right  direc- 
tion, and  particularly  rich  in  its  examples  for  practice  and  drill.  It  seems  too 
to  be  very  carefully  graded.  I  do  not  see  how  one  can  fail  to  understand  the 
elements  of  algebra  if  he  is  held  up  to  the  plan  of  this  book." 

— Principal  F.  A.  Hill,  Cambridge,  Mass. 

ELEMENTARY  ALGEBRA. 
By  J.  Hamblin  Smith,  of  Gonvilleand  Caius  College,  Cambridge.      i2mo. 
80  cents.    New  and  Revised  Edition  (1894).     i2mo.     416  pages.    $i.co. 

In  this  edition  the  whole  of  the  old  book  has  been  retained  and  many  addi- 
tions have  been  made,  including  over  1,000  new  Examples.  A  chapter  contain- 
ing Kxplanations,  the  Remainder.  Theorem,  Symmetry,  Cyclical  Order,  etc., 
and  a  new  chapter  on  the  Exponential  1  heorem.  Logarithms,  etc. 

ELEMENTARY  ALGEBRA.     With  Numerous  Examples. 
By  W.  A.  Potts,  B.A.,  and  W.   L.  Sargant,  B.A.      i2mo.     146  pages. 
60  cents. 

LONGMANS,  GREEN,  &  CO.,  91-93  Fifth  Avenue,  New  York. 


LONGMANS,  GREEN,  ^  CO:S  PUBLICATIONS, 

ELEMENTARY   AND    CONSTRUCTIVE   GEOMETRY. 

By  Edgar  H.  Nichols,  A.B  ,  of  the  Browne  and  Nichols  School,  Cam- 
bridge, Mass.  Crown  8vo.  150  pages.  With  loi  figures.  75  cents. 
This  book  is  prepared  mainly  with  reference  to  the  recommendations  of  the 
National  Committee  of  Ten  which  are  as  follows  :  "  At  about  the  age  of  ten  for 
the  average  child,  systematic  instruction  in  concrete  or  experimental  geometry 
should  begin,  and  should  occupy  about  one  school  hour  per  week  for  at  least 
three  years.  During  this  period  the  main  facts  of  plane  and  solid  geometry 
should  be  taught,  not  as  an  exercise  in  logical  deduction  and  exact  demonstra- 
tion, but  in  as  concrete  and  objective  a  form  as  possible.  For  example,  the 
simple  properties  of  similar  plane  figures  and  similar  soHds  should  not  be 
proved,  but  should  be  illustrated  and  confirmed  by  cutting  up  and  re-arrang- 
ing drawings  or  models. 

AN  INTRODUCTORY  COURSE  IN  DIFFERENTIAL  EQUA- 
TIONS.     For  Students  in  Classical  and  Engineering  Colleges. 

By  D.  A.  Murray,  A.B.,  Ph.D.,  formerly  Scholar  and  Fellow  of 
Johns  Hopkins  University  ;  Instructor  in  Mathematics  at  Cornell 
University.     Crown  8vo.    Over  250  pages.     $1.90.*  \In  Press. 

A  TREATISE  ON  COMPUTATION.  An  Account  of  the  Chief 
Methods  for  Contracting  and  Abbreviating  Arithmetical  Calcu- 
lations. 

By  Edv^ard  M.  Langley,  M.A.     i2mo.     pp.  viii-184.     $1.00. 

GRAPHICAL   CALCULUS. 

By  Arthur  H.  Barker,  B.A.,  B.Sc,  Senior  Whitworth  Scholar  1895. 

With  an  Introduction   by  John  Goodman,  A.M.I.C.E.,  Professor 

of   Engineering   at   the  Yorkshire   College,   Victoria    University. 

Crown  8vo.     197  pages.     $1.50. 

"  This  is  an  able  mathematical  work,  written  with  refreshing  originality.     It 

forms  unquestionably  the  best  introduction  to  the  study  of  the  Calculus  that  we 

have  yet  seen.     Never  once  does  Mr.  Barker  leave  his  reader  in  doubt  as  to  the 

significance  of  the  result  obtained.     His  appeal  to  the  graph  is  conclusive,  and 

so  unfolds  the  mysteries  of  the  Calculus  that  the  delighted  student  grasps  the 

utility  of  the  subject  from  the  sXaxV — Schoolmaster . 

ALGEBRA  FOR  SCHOOLS  AND  COLLEGES. 

By  William  Freeland,  A.B.,  Head  Master  at  the  Harvard  School, 
New  York  City.     i2mo.     320  pages.     $1.40. 

••  Excellent  features  of  the  book  may  be  found  in  the  adequate  treatment  of 
factoring,  in  the  discussion  of  the  theory  of  exponents,  and  in  the  chapter  of 
radicals.  The  form  of  the  demonstration  of  processes  is  here  exceptionally  fine, 
the  whole  showing  a  nice  sense  of  logic  which  'looks  at  the  end  from  the  begin- 
ning.' It  would  be  difficult  to  find  a  book  cr  ntaining  a  better  selection  of  exer- 
cises, or  one  in  which  the  publishers  have  more  carefully  interpreted  the  author's 
ideas  of  arrangement." — Educational  Review. 

Note.— The  Answers  to  the  Examples  in  this  book  are  printed  in  a  separate 
pamphlet  and  will  be  furnished,  free  of  charge,  only  to  Teachers  using  the 
book,  or  to  students  upon  the  written  request  of  such  Teachers. 


LONGMANS,  GREEN,  &  CO.,  91-93  Fifth  Avenue,  New  York. 


^m^fmfmmm 


Yb  bbtt4 


6stUl 
18364? 


